Kyushu J. Math.
Vol. 58, 2004, pp. 167–202
SUBMANIFOLD GEOMETRIES IN A
SYMMETRIC SPACE OF NON-COMPACT TYPE
AND A PSEUDO-HILBERT SPACE
Naoyuki KOIKE
(Received 10 May 2003)
0. Introduction
It is very useful to find methods to replace the investigation of submanifolds in a
curved space (i.e. a non-flat Riemannian manifold) with that of submanifolds in a flat
space (i.e. a Euclidean space or a Hilbert space). In [T2] and [TT], Terng and
Thorbergsson introduced such a method. For a symmetric space G/K of compact
type, they defined one Riemannian submersion of a Hilbert space onto G/K as
follows. Let g be the Lie algebra of G and ·, · be the Ad(G)-invariant inner product
of g inducing the Riemannian metric of G/K. Let H 0 ([0, 1], g) be the Hilbert space
paths u : [0, 1] → g, where the L2 -inner product is defined by
of all L2 -integrable
1
u, v0 := 0 u(t), v(t) dt. Also, let H 1 ([0, 1], g) be the space of all absolutely
continuous paths u : [0, 1] → g such that u ∈ H 0 ([0, 1], g) and let H 1 ([0, 1], G) be
the Hilbert Lie group of all absolutely continuous paths g : [0, 1] → G such that g is
square integrable, that is, g∗−1 g ∈ H 0 ([0, 1], g), where g is the velocity vector field
0
of g and g∗−1 g is the path t → L−1
g(t )∗ g (t) in g. Define a map φ : H ([0, 1], g) → G
0
by φ(u) = gu (1) (u ∈ H ([0, 1], g)), where gu is the element of H 1 ([0, 1], G)
−1 g = u. This map φ
satisfying gu (0) = e (e is the identity element of G) and gu∗
u
is called the parallel transport map (from 0 to 1). Denote by π the natural projection
of G onto G/K. They showed that φ (and hence π ◦ φ) becomes a Riemannian
be one component
submersion. Let M be an immersed submanifold in G/K and M
−1
of (π ◦ φ) (M). They investigated the relation between the submanifolds M and M.
More precisely, they showed the following facts:
is a proper Fredholm submanifold;
(i) if M is compact, then M
is isoparametric
(ii) M is equifocal (respectively weakly equifocal) if and only if M
(respectively weakly isoparametric).
See [T2] and [TT] about the definitions of a proper Fredholm submanifold and
an equifocal (or weakly equifocal) submanifold.
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N. Koike
Remark 0.1. According to the proof (the proof of Lemma 5.8 in [TT]) of the fact (i),
becomes a proper Fredholm submanifold for a properly immersed
it is shown that M
submanifold M in general.
The author [Ko3] recently investigated the spectrum of the shape operator of
(π ◦ φ)−1 (M) for a curvature adapted submanifold M in a symmetric space G/K
of compact type, where a curvature adapted submanifold is a notion defined by Berndt
and Vanhecke in [BV]. Here we note that all submanifolds in a real space form, all
Kaehlerian (or generic) submanifolds in a complex space form and the principal orbits
of the isotropy representation of an arbitrary symmetric space are curvature adapted.
Also, it is known that constant tubes over a curvature adapted submanifold are again
curvature adapted (see [BV]). As its application, the author showed that if M is a
minimal and curvature adapted submanifold in a symmetric space of compact type,
then (π ◦ φ)−1 (M) is a formal minimal submanifold in H 0 ([0, 1], g). See [Ko3]
about the definition of formal minimality of a proper Fredholm submanifold. Also, he
showed that if M is an isospectral and curvature adapted submanifold in a rank-one
symmetric space of compact type, then (π ◦ φ)−1 (M) is an isospectral submanifold in
H 0 ([0, 1], g), where an isospectral submanifold implies that the shape operators Av1
and Av2 have the same spectrum considering multiplicities for arbitrary unit normal
vectors v1 and v2 . For example, focal submanifolds of isoparametric hypersurfaces
are isospectral.
In [TT], Terng and Thorbergsson proposed the following problem:
Is there similar theory for equifocal submanifolds in a symmetric space
of non-compact type?
In this paper, we tackle this problem. In the case where G/K is a symmetric
space of non-compact type, the Ad(G)-invariant non-degenerate symmetric bilinear
form ·, · of g inducing the Riemannian metric of G/K is not positive definite. Hence
the above Hilbert space H 0 ([0, 1], g) cannot be defined. In this paper, we define
the space corresponding to H 0 ([0, 1], g) and further a submersion corresponding to
the above φ. We denote the corresponding space (respectively the corresponding
submersion) by the same symbol H 0 ([0, 1], g) (respectively φ). In the sense of this
paper, the space H 0 ([0, 1], g) becomes a pseudo-Hilbert space; also, it is shown
that the submersion π ◦ φ becomes a pseudo-Riemannian submersion and, for a
submanifold M in G/K, we show that (π ◦φ)−1 (M) becomes a Fredholm submanifold
in H 0 ([0, 1], g). Also, we define the notion of the complex equifocality of a
submanifold in a symmetric space of non-compact type and those of the complex
Submanifold geometries in a symmetric space
169
(or proper complex) isoparametricness and real (or proper real) isoparametricness of
a Fredholm submanifold in a pseudo-Hilbert space (see Sections 2 and 3). In the
case where M is a curvature adapted submanifold, we describe the spectrum of a
(complexified) shape operator of the Fredholm submanifold (π ◦ φ)−1 (M) in terms
of that of a shape operator of M and (restricted) roots of G/K (see Theorem 5.8).
In terms of those descriptions, we obtain the following result for the complex (or
proper complex) isoparametricness of each component of the Fredholm submanifold
(π ◦ φ)−1 (M).
T HEOREM A. Let M be an immersed curvature adapted submanifold with globally
flat and abelian normal bundle in a simply connected symmetric space N = G/K of
non-compact type, φ be the parallel transport map for G, π be the natural projection
be one component of (π ◦ φ)−1 (M). Then the following
of G onto G/K and M
statements hold:
is complex isoparametric;
(i) M is complex equifocal if and only if M
(ii) M is complex equifocal and for each v ∈ T ⊥ M, ±α(g∗−1 v)’s (α ∈ + \ g −1 v )
∗
is proper complex
are not principal curvatures of direction v if and only if M
isoparametric, where g is a representative element of the base point of v, + is
the positive root system with respect to a maximal abelian subspace containing
g∗−1 v and g −1 v := {α ∈ + | α(g∗−1 v) = 0}.
∗
Also, we obtain the following result for the real (or weakly real) isoparametricness
of each component of the Fredholm submanifold (π ◦ φ)−1 (M).
T HEOREM B. Let M be an immersed submanifold with globally flat and abelian
normal bundle in a simply connected symmetric space N = G/K of non-compact
be one component of (π ◦ φ)−1 (M), where φ and π are as in
type and M
Theorem A. Then M is equifocal (respectively weakly equifocal) if and only if M
is real isoparametric (respectively weakly real isoparametric).
Also, we obtain the following result for the formal minimality of each component
of the Fredholm submanifold (π ◦ φ)−1 (M).
T HEOREM C. Let M be an immersed curvature adapted submanifold in a symmetric
be one component of (π ◦ φ)−1 (M),
space N = G/K of non-compact type and M
where φ and π are as in Theorem A. Assume that, for each v ∈ T ⊥ M, ±α(g∗−1 v)’s
(α ∈ + \ g −1 v ) are not principal curvatures of direction v, where g is a repre∗
sentative element of the base point of v, and + and g −1 v are as in Theorem A. Then
∗
is formally extremal.
M is minimal if and only if M
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N. Koike
Furthermore, we obtain the following result for the isospectrality of each
component of the Fredholm submanifold (π ◦ φ)−1 (M).
T HEOREM D. Let M be an immersed curvature adapted submanifold in a rank be one component of
one symmetric space N = G/K of non-compact type and M
−1
(π ◦ φ) (M), where φ and π are as in Theorem A. If M is isospectral, then so is M.
In Section 1, we prepare basic notions and facts. In Section 2, we introduce
the new notions of a pseudo-Hilbert space, a pseudo-Riemannian Hilbert manifold,
a Fredholm (pseudo-Riemannian Hilbert) submanifold, a real (or proper real)
isoparametric submanifold, a complex (or proper complex) isoparametric submanifold
and a formally extremal Fredholm submanifold. In Section 3, we introduce the
new notions of a complex focal radius and the (proper) complex equifocality of a
submanifold in a symmetric space of non-compact type. In Section 4, we define the
parallel transport map for a semi-simple non-compact Lie group by imitating the case
of a compact Lie group. We also show that the map becomes a pseudo-Riemannian
submersion. In Section 5, we describe the spectrum of a (complexified) shape operator
in terms of that of a shape operator of M and
of the above Fredholm submanifold M
(restricted) roots of G/K. In Section 6, we prove Theorems A–D. In Section 7, we
define a certain kind of affine transformation group for a proper complex isoparametric
submanifold.
Throughout this paper, we assume that all geometric objects are of class C ∞ and
all manifolds are connected ones without boundary.
1. Basic notions and facts
In this section, we recall basic notions and facts. First we recall some classes of
submanifolds in a symmetric space. Let N = G/K be a symmetric space, (g, σ ) be its
orthogonal symmetric Lie algebra and p be the eigenspace for −1 of σ . The subspace
p is identified with the tangent space TeK N of N at eK, where e is the identity element
of G. Let h be a maximal abelian subspace of p. For each C-valued linear function
α (= 0) on h, set pα := {X ∈ p | ad(a)2 (X) = α(a)2 X for all a ∈ h}, where ad
is the adjoint representation of g. If pα = {0}, then the linear function α is called
a (restricted) root for h and pα is called the root space for α. For convenience,
we sometimes interpret h as p0 . Note that, if N is of compact type (respectively
of non-compact type), then all roots are purely imaginary (respectively real) valued.
The set of all roots for h is called the root system for h. Let + be the set of
all positive roots with respect to some lexicographic ordering of h. Then we have
Submanifold geometries in a symmetric space
171
p = h + α∈+ pα , which is called the root space decomposition for h. Let M
be an immersed submanifold in N and T ⊥ M be its normal bundle. If, for each
x(= gK) ∈ M, g∗−1 Tx⊥ M is an abelian subspace in p, then M is said to have abelian
normal bundle. Also, if the normal connection of M is flat and has trivial holonomy,
then M is said to have globally flat normal bundle. If, for each v ∈ T ⊥ M, the operator
R(·, v)v leaves Tx M invariant (x is the base point of v) and it is commutative with the
shape operator Av , then M is said to be curvature adapted.
Next we recall the definitions of focal points and focal radii of a pseudoRiemannian submanifold M in a pseudo-Riemannian manifold N. Denote by exp⊥
the normal exponential map of M. Denote by U ⊥ M the unit normal bundle of M.
⊥
Let v ∈ Ux⊥ M. If dim Ker exp⊥
∗rv = ν(= 0), then exp (rv) (respectively r) is called
a focal point (respectively a focal radius) with multiplicity ν of M along the normal
⊥
geodesic exp⊥ tv, where exp⊥
∗rv is the differential of exp at rv.
Let M be an immersed submanifold with globally flat and abelian normal bundle
in a (Riemannian) symmetric space N. Let ṽ be a parallel unit normal vector field
of M. Assume that the number (which may be zero or infinite) of focal radii
along the geodesic γṽx is independent of the choice of x ∈ M. Let {r1,x , r2,x , . . . }
(r1,x < r2,x < · · · ) be the set of all focal radii of ṽx . Let ri (i = 1, 2, . . . ) be functions
on M defined by assigning ri,x to each x ∈ M. These functions ri (i = 1, 2, . . . )
are called focal radius functions for ṽ. The normal vector field ri ṽ is called a focal
normal vector field for ṽ. If M is compact and, for each parallel unit normal vector
field ṽ of M, the number of the focal radii of ṽx is independent of the choice of
x ∈ M and further each focal radius function for ṽ is constant on M (respectively
along the corresponding focal leaves), then M is called an equifocal (respectively
weakly equifocal) submanifold. These notions were introduced in [TT]. We use those
terminologies without the compactness of M. Similarly, we define the equifocality and
the weakly equifocality of a pseudo-Riemannian submanifold in a pseudo-Riemannian
symmetric space.
2. Pseudo-Hilbert space and pseudo-Riemannian Hilbert manifolds
In this section, we introduce new notions of a pseudo-Hilbert space, a pseudoRiemannian Hilbert manifold, a Fredholm pseudo-Riemannian Hilbert submanifold,
a real (or proper real) isoparametric submanifold and a complex (or proper complex)
isoparametric submanifold. Let V be an infinite-dimensional real vector topological
space and ·, · be a non-degenerate symmetric bilinear form of V such that ·, ·
(: V × V → R) is continuous with respect to the topology of V . Let V = V− ⊕ V+
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N. Koike
be an orthogonal decomposition of V such that ·, ·|V− ×V− is negative definite and
·, ·|V+ ×V+ is positive definite. We call such a decomposition an orthogonal timespace decompositon. Set ·, ·V± := −πV∗− ·, · + πV∗+ ·, ·, where πV− (respectively
πV+ ) is the projection of V onto V− (respectively V+ ). It is clear that ·, ·V± is
positive definite, that is, an inner product of V . If there exists an orthogonal time-space
decomposition V = V− ⊕ V+ such that (V , ·, ·V± ) is a Hilbert space (respectively
a separable Hilbert space) and that the distance topology associated with ·, ·V±
coincides with the original topology of V , then we call (V , ·, ·) a pseudo-Hilbert
space (respectively a separable pseudo-Hilbert space). In the following, assume
that all pseudo-Hilbert space are separable. Let {vi }∞
i=1 be a linearly independent
system of a pseudo-Hilbert space (V , ·, ·) such that, for each i ∈ N, there exists
î ∈ N satisfying |vi , vj | = δîj (j ∈ N), where δ·· is the Kronecker symbol.
We call such a system a pseudo-orthonormal system of (V , ·, ·). In particular,
if î = i for all i ∈ N, then we call such a system an orthonormal system of
(V , ·, ·). Furthermore, if a pseudo-orthonormal (respectively orthonormal) system
∞
∞
{vi }∞
i=1 Span{vi } = V , then we call {vi }i=1 a pseudo-orthonormal
i=1 satisfies
(respectively orthonormal) base of (V , ·, ·), where · is the closure of · with respect
to the original topology of V . Denote by ·, ·H the non-degenerate Hermitian bilinear
form of the complexification V c of V defined naturally from ·, ·. Similarly, we
define a pseudo-orthonormal system (and base) and an orthonormal system (and base)
of (V c , ·, ·H ).
L EMMA 2.1. A pseudo-orthonormal base of (V , ·, ·) or (V c , ·, ·H ) is a maximal
pseudo-orthonormal system.
∞
Proof. Let {ei }∞
i=1 be a pseudo-orthonormal base of (V , ·, ·). Suppose that {ei }i=1 is
not a maximal pseudo-orthonormal system, that is, there exists a pseudo-orthonormal
∞
system {ei }∞
i=1 ∪ E which properly contains {ei }i=1 . Let e0 ∈ E. From the
definition of a pseudo-orthonormal system, we have e0 , ei = 0 for all i ∈ N.
∞
∞
Since e0 ∈ V =
i=1 Span{ei }, we can write e0 =
i=1 ai ei . Then, from the
continuity of ·, ·, we have e0 , eî = ai ei , eî , that is, ai = 0 for all i ∈ N. Thus
we have e0 = 0. This contradicts the fact that {ei }∞
i=1 ∪ E is a pseudo-orthonormal
is
a
maximal
pseudo-orthonormal
system. Similarly, the
system. Therefore, {ei }∞
i=1
2
maximality of a pseudo-orthonormal base of (V c , ·, ·H ) is also shown.
We now define a notion of a pseudo-Riemannian Hilbert manifold. Let M be
a C k -Hilbert manifold (k ≥ 1) modelled on a separable Hilbert space (V , ·, ·V ).
Note that each tangent space Tx M is identified with V and hence it has the distance
topology associated with ·, ·V . Let ·, · be a C k -section of the tensor bundle
Submanifold geometries in a symmetric space
173
T ∗ M ⊗ T ∗ M such that ·, ·x is a continuous non-degenerate symmetric bilinear form
on Tx M for each x ∈ M. If, for each x ∈ M, there exist C k -distributions W± on
some neighborhood U of x such that W±y (y ∈ U ) give an orthogonal time-space
decomposition of (Ty M, ·, ·y ) and that (Ty M, ·, ·y,W±y ) (y ∈ U ) are isometric
to (V , ·, ·V ), then we call (M, ·, ·) a C k -pseudo-Riemannian Hilbert manifold,
∗ ·, · + π ∗ ·, · (π
where ·, ·y,W±y = −πW
y
y
W±y is the projection of Ty M
W+y
−y
onto W±y ). Pseudo-Hilbert spaces are regarded as C ω -pseudo-Riemannian Hilbert
manifolds. Let f be a C k -immersion of a C k -Hilbert manifold M into a pseudoHilbert space (V , ·, ·) (k ≥ 1). If (M, f ∗ ·, ·) is a C k−1 -pseudo-Riemannian
Hilbert manifold, then we call M a C k−1 -pseudo-Riemannian Hilbert submanifold
in (V , ·, ·) immersed by f . In the following, assume k = ∞. Let ∇ (respectively ∇)
be the Levi–Civita connection of f ∗ ·, · (respectively ·, ·). The shape tensor A and
the normal connection ∇ ⊥ of M are usually defined by
X v = −f∗ (Av X) + ∇ ⊥ v
∇
X
(X ∈ T M, v ∈ (T ⊥ M)),
⊥ v ∈ T ⊥ M. Note that A is not necessarily diagonalizable
where Av X ∈ T M and ∇X
v
with respect to an orthonormal base because f ∗ ·, · is not necessarily positive
definite. We call M a Fredholm pseudo-Riemannian Hilbert submanifold (or simply a
Fredholm submanifold) if the following conditions hold:
(F-i)
M is of finite codimension;
(F-ii)
there exists an orthogonal time-space decomposition V = V− ⊕V+ such that
(V , ·, ·V± ) is a Hilbert space and that, for each v ∈ T ⊥ M, Av is a compact
operator with respect to f ∗ ·, ·V± .
Since Av is a compact operator with respect to f ∗ ·, ·V± , for each v ∈ T ⊥ M,
the operator id −Av is a Fredholm operator with respect to f ∗ ·, ·V± and hence the
normal exponential map exp⊥ : T ⊥ M → V of M is a Fredholm map with respect
to the metric of T ⊥ M naturally defined from f ∗ ·, ·V± and ·, ·V± , where id is
the identity transformation of T M. The set of all eigenvalues of Av is described as
{0} ∪ {λi | i = 1, 2, . . . }, where |λi | > |λi+1 | or λi = −λi+1 > 0. Also, the set of all
eigenvalues of the complexification Acv of Av is described as {0} ∪ {µi | i = 1, 2, . . . },
where ‘|µi | > |µi+1 |’ or ‘|µi | = |µi+1 | and Re µi > Re µi+1 ’ or ‘|µi | = |µi+1 |
and Re µi = Re µi+1 and Im µi = −Im µi+1 > 0’. We call λi (respectively µi )
the ith real (respectively complex) principal curvature of direction v. Assume that
M has globally flat normal bundle. Fix a parallel normal vector field ṽ on M.
Assume that the number (which may be infinite) of distinct real (respectively complex)
principal curvatures of ṽx is independent of x ∈ M. Then we can define functions
λ̃i (respectively µ̃i ) (i = 1, 2, . . . ) on M by assigning the ith real (respectively
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N. Koike
complex) principal curvature of direction ṽx to each x ∈ M. We call this function
λ̃i (respectively µ̃i ) the ith real (respectively complex) principal curvature function of
direction ṽ. If the number of distinct real (respectively complex) principal curvatures
is finite, then each λ̃i (respectively µ̃i ) has constant multiplicity. However, if the
number is infinite, then each λ̃i and µ̃i do not necessarily have constant multiplicity.
Let M be a Fredholm submanifold with globally flat normal bundle satisfying the
following condition:
(RI-i)
for each parallel normal vector field ṽ, the number of distinct real principal
curvatures of direction ṽx is independent of the choice of x ∈ M.
We call M a real isoparametric (respectively a weakly real isoparametric)
submanifold if the following further condition (RI-ii) (respectively (WRI-ii)) holds:
(RI-ii) for each parallel normal vector field ṽ, each real principal curvature function
of direction ṽ is constant on M;
(WRI-ii) for each parallel normal vector field ṽ, each real principal curvature function
of direction ṽ is constant along leaves of the principal foliation for the real
principal curvature function and has constant multiplicity.
Let M be a Fredholm submanifold with globally flat normal bundle satisfying the
following condition:
(CI)
for each parallel normal vector field ṽ, the number of distinct complex
principal curvatures of direction ṽx is independent of the choice of x ∈ M
and each complex principal curvature function of direction ṽ is constant
on M.
We call such a submanifold M a complex isoparametric submanifold.
Remark 2.1.
(i) Each real (respectively complex) principal curvature function of a real
(respectively complex) isoparametric submanifold has automatically constant
multiplicity.
(ii) In the case where the ambient space is of finite dimension, real (or complex)
isoparametric submanifolds are not necessarily isoparametric in the sense of
[Ko1].
Let M be a real isoparametric submanifold in a pseudo-Hilbert space (V , ·, ·).
If, for each v ∈ T ⊥ M, there exists an orthonormal tangent base consisting of the
eigenvectors of the shape operator Av , then we call M a proper real isoparametric
submanifold. Then, for each x ∈ M, there exists an orthonormal tangent base
consisting of the common-eigenvectors of the shape operators Av (v ∈ Tx⊥ M) because
Av are commutative. Let {Ei | i ∈ I } (I ⊂ N) be the family of distributions on M
such that, for each x ∈ M, {Ei (x) | i ∈ I } are the set of all common-eigenspaces
Submanifold geometries in a symmetric space
175
of Av (v ∈ Tx⊥ M). Note that Tx M =
i∈I Ei (x) holds, where · implies the
closure of ·. There exist smooth sections λi (i ∈ I ) of the dual bundle (T ⊥ M)∗
of T ⊥ M such that Av = λi (v) id on Ei (π(v)) for each v ∈ T ⊥ M, where π is the
bundle projection of T ⊥ M. We call λi (i ∈ I ) principal curvatures of M and call
subbundles Ei (i ∈ I ) of T ⊥ M curvature distributions of M. Note that λi (v) is one
of the principal curvatures of direction v. Define a normal vector field vi (i ∈ I ) by
λi (·) = vi , ·. Note that each vi is parallel with respect to ∇ ⊥ . We call vi (i ∈ I )
curvature normals of M.
Remark 2.2. In the case where the ambient space is of finite dimension, the proper real
isoparametricness coincides with the proper isoparametricness of the sense of [Ko1].
Let M be a complex isoparametric submanifold in a pseudo-Hilbert space
(V , ·, ·). If, for each v ∈ T ⊥ M, there exists a pseudo-orthonormal base of Tx M c (x :
the base point of v) consisting of the eigenvectors of the complexified shape operator
Acv , then we call M a proper complex isoparametric submanifold. Then, for each
x ∈ M, there exists a pseudo-orthonormal base of Tx M c consisting of the commoneigenvectors of the complexified shape operators Acv (v ∈ Tx⊥ M) because Acv are
commutative. Let {Fi | i ∈ I } (I ⊂ N) be the family of subbundles of T M c such
that, for each x ∈ M, {Fi (x) | i ∈ I } is the set of all common-eigenspaces of Acv
(v ∈ Tx⊥ M). Note that Tx M c = i∈I Fi (x) holds. There exist smooth sections µi
(i ∈ I ) of (T ⊥ M c )∗ such that Acv = µi (v) id on Fi (π(v)) for each v ∈ T ⊥ M, where
π is the bundle projection of T ⊥ M c . We call µi (i ∈ I ) complex principal curvatures
of M and call subbundles Fi (i ∈ I ) of T ⊥ M c complex curvature distributions
of M. Note that µi (v) is one of the complex principal curvatures of direction v.
Define a complex normal vector field wi (i ∈ I ) by µi (·) = wi , ·c , where ·, ·c
is the complexification of ·, ·. Note that each wi is parallel with respect to the
complexification ∇ ⊥c of ∇ ⊥ . We call wi (i ∈ I ) complex curvature normals of M.
Remark 2.3. Let (V , ·, ·) be a pseudo-Hilbert space and ·, ·H be the non-degenerate
Hermitian bilinear form of V c defined naturally from ·, ·. If A is a Hermitian
transformation of (V c , ·, ·H ), that is, AX, Y H = X, AY H (X, Y ∈ V c ) and λ
is a non-real eigenvalue of A, then, for the eigenspace Dλ of λ, ·, ·H |Dλ ×Dλ = 0
holds. Also, if λ and µ are eigenvalues of A with λ̄ = µ, then ·, ·H |Dλ ×Dµ = 0
holds, where λ̄ is the conjugate of λ. Hence, if λ is a non-real eigenvalue of A, then λ̄
is also an eigenvalue of A with the same multiplicity as λ.
In [Ko3], we defined the formally minimality of a proper Fredholm submanifold
in a Hilbert space. By imitating the definition, we define the formally extremality of
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N. Koike
a Fredholm submanifold in a pseudo-Hilbert space as follows. Let M be a Fredholm
submanifold in a pseudo-Hilbert space (V , ·, ·). Denote by A the shape tensor of M.
Let v be a unit normal vector of M and let Spec Acv = {0} ∪ {µi | i = 1, 2, . . . },
where ‘|µi | > |µi+1 |’ or ‘|µi | = |µi+1 | and Re µi > Re µi+1 ’ or ‘|µi | = |µi+1 |
and Re µi = Re µi+1 and Im µi = −Im µi+1 > 0’. Here, if Spec Acv is finite, then
we promise µi = 0 (i ≥ Spec Acv ), where Spec Acv is the number of elements of
Spec Acv . Let mi be the multiplicity of µi (= 0). Assume that there exists a pseudoorthonormal base of Tx M c (x is the base point of v) consisting of eigenvectors of Acv
and ∞
m µ exists, where we promise mi = 0 when µi = 0. Then we call the sum
∞ i=1 i i
c
c
m
µ
i
i the trace of Av and denote it by Tr Av .
i=1
Definition. If, for each unit normal vector v of M, Tr Acv = 0 holds, then we call M a
formally extremal Fredholm submanifold.
Remark 2.4. King and Terng [KT] defined the notion of the ζ -trace Trζ A for a
compact and self-adjoint operator A of a Hilbert space V as follows:
∞
∞
+ + s
− − s
mi (λi ) −
mi |λi | ,
Trζ A = lim
s→1+0
i=1
i=1
−
+
+
where Spec A = {0} ∪ {λ+
i | i = 1, 2, . . . } ∪ {λi | i = 1, 2, . . . } (λi > λi+1 > 0 >
−
−
±
±
λi+1 > λi (i = 1, 2, . . . )) and mi is the multiplicity of λi . On the other hand, we
[Ko3] thought that the trace of A should be defined as follows:
Tr A =
∞
mi λi ,
(2.1)
i=1
where Spec A = {0} ∪ {λi | i = 1, 2, . . . } (|λi | > |λi+1 | or λi = −λi+1 > 0
(i = 1, 2, . . . )) and mi is the multiplicity of λi . How we take such a trace is similar to
the Fourier expansion:
v = lim
v, ei ei (v ∈ V ),
k→∞
i∈Ik
where {ei }∞
i=1 is an orthonormal base of V and Ik = {i | |v, ei | ≥ 1/k} (k =
1, 2, . . . ). Hence the trace (2.1) and the above Tr Acv seem to be natural definitions.
Here we give an example of A satisfying Trζ A = Tr A. If
∞
(−1)k+1 (−1)k
Spec A =
∪
k = 1, 2, . . . ,
√
√
k
k k=1
Submanifold geometries in a symmetric space
177
then we have Tr A = 0 and
∞
∞
∞
1
1
(−1)k s
Trζ A = lim
−
−
√
,
√
√
s
s
s→1+0
k
2k + 1
2k
k=0
k=0
k=1
which does not exist.
At the end of this section, we define the notion of a pseudo-Riemannian
submersion of one pseudo-Riemannian Hilbert manifold onto another pseudoRiemannian Hilbert manifold. Let M and N be pseudo-Riemannian Hilbert manifolds
and π : M → N be a submersion. Assume that fibers π −1 (x) (x ∈ N) are pseudoRiemannian Hilbert submanifolds in M. Furthermore, assume that the restriction
π∗ |Hy of the differential π∗ to the horizontal subspace Hy (= Ty⊥ π −1 (π(y))) is a
linear isometry for each y ∈ M. Then we call π a pseudo-Riemannian submersion of
M onto N.
3. Complex focal radii and the complex equifocality
For a submanifold in a hyperbolic space H m (c) of constant curvature c, there does
not exist a focal radius corresponding to a principal curvature whose absolute value
√
is smaller than or equal to −c. This fact indicates that an imaginary focal radius
should be defined for submanifolds in a complete Riemannian manifold of negative
sectional curvature. In this section, we define imaginary focal radii for submanifolds
in a symmetric space of non-compact type. Let M be an immersed submanifold in
a symmetric space N = G/K of non-compact type. Denote by A the shape tensor
of M. Let v ∈ Tx⊥ M and X ∈ Tx M (x = gK). Denote by γv the geodesic in N with
γ̇v (0) = v, where γ̇v (0) is the velocity vector of γv at 0. The Jacobi field J along γv
with J (0) = X and J (0) = −Av X is given by
co
si
J (s) = (Pγv |[0,s] ◦ (Dsv
− sDsv
◦ Av ))(X),
v J , Pγv |
where J (0) = ∇
[0,s] is the parallel translation along γv |[0,s] ,
√
co
= g∗ ◦ cos( −1 ad(sg∗−1 v)) ◦ g∗−1
Dsv
and
si
Dsv
√
sin( −1 ad(sg∗−1 v))
= g∗ ◦ √
◦ g∗−1
−1 ad(sg∗−1 v)
(see [TT] or [Ko2] in detail). Here ad is the adjoint representation of the Lie
algebra g of G. Assume that M has abelian normal bundle. A focal radius of
178
N. Koike
M along γv is catched as a real number s0 with Ker(Dsco0 v − s0 Dssi0 v ◦ Av ) = {0}.
So, we call a complex number z0 with Ker(Dzco0 v − z0 Dzsi0 v ◦ Acv ) = {0} a
complex focal radius of M along γv and call dim Ker(Dzco0 v − z0 Dzsi0 v ◦ Acv ) the
multiplicity of the complex focal radius z0 , where Dzco0 v (respectively Dzsi0 v ) implies
√
the complexification
of a map √
(g∗ ◦ cos( −1 ad(z0 g∗−1 v)) ◦ g∗−1 )|Tx M (respectively
√
(g∗ ◦ [sin( −1 ad(z0 g∗−1 v))/ −1 ad(z0 g∗−1 v)] ◦ g∗−1 )|Tx M ) from Tx M to Tx N c .
c
Also, for a complex focal radius z0 of M along γv , we call z0 v (∈ Tx⊥ M ) a complex
focal normal of M at x.
P ROPOSITION 3.1. Let Av X = λX and X ∈ g∗ pα , where g is the representative
element of the base point of v and pα is the root space for a root α with respect to a
maximal abelian subspace h containing g∗−1 v.
(i) If |λ| > |α(g∗−1 v)| = 0, then 1/λ is a focal radius along γv , where we admit the
case of α = 0 (i.e. pα = h).
√
(ii) If |λ| > |α(g∗−1 v)| > 0, then (1/α(g∗−1 v))(arctanh[α(g∗−1 v)/λ] + j π −1)
(j ∈ Z) are complex focal radii along γv .
√
(iii) If |λ| < |α(g∗−1 v)|, then (1/α(g∗−1 v))(arctanh[λ/α(g∗−1 v)] + (j + 12 )π −1)
(j ∈ Z) are complex focal radii along γv .
co − zD si ◦ Ac )(X) = (1 − zλ)X. Hence 1/λ
Proof. If α(g∗−1 v) = 0, then we have (Dzv
zv
v
is a focal radius along γv . Therefore, statement (i) is deduced. Assume α(g∗−1 v) = 0.
Since
√
√
λ sin( −1α(g∗−1 v)z)
co
si
c
−1
X,
(Dzv − zDzv ◦ Av )(X) = cos( −1α(g∗ v)z) −
√
α(g∗−1 v) −1
√
√
the complex numbers (1/α(g∗−1 v) −1) arctan(α(g∗−1 v) −1/λ) are complex focal
radii along γv . On the other hand, we have
√
α(g∗−1 v) −1
arctan
 λ
α(g∗−1 v)
√


+ jπ
(j = 0, ±1, . . . )
−1 arctanh


λ



(when |λ| > |α(g∗−1 v)|),
= √
1
λ


+ j+
−1 arctanh
π (j = 0, ±1, . . . )

−1

2

α(g∗ v)



(when |λ| < |α(g∗−1 v)|).
Hence we obtain the statements (ii) and (iii).
2
Submanifold geometries in a symmetric space
179
Now we describe the situation of complex focal radii of totally umbilic
hypersurfaces in a hyperbolic space.
Example. Let Mλ be a totally umbilical hypersurface in the m-dimensional hyperbolic
space H m (c) of constant curvature c having the principal curvature λ (≥0), where
√
we fix a unit normal vector field v of Mλ . Assume that λ = −c. According to
Proposition 3.1, the set F of all the complex focal radii along γv is given by
√

√
√
1
−c

 √
arctanh
+
j
π
−1
|
j
=
0,
±1,
.
.
.
(λ > −c),


λ

 −c √
F =
λ
1
1

π
arctanh
+
j
+
−1
|
j
=
0,
±1,
.
.
.
√
√


2
−c
−c


√

(0 ≤ λ < −c).
Set
rλ,j

1



 √−c
:=

1


√
−c
√
√
√
−c
+ j π −1
(λ > −c),
λ
√
√
λ
1
(0 ≤ λ < −c)
arctanh √
+ j+
π −1
2
−c
arctanh
(j = 0, ±1, ±2, . . . ). These complex focal radii rλ,j continuously variate on the onepoint compactification C ∪ {∞}(≈S 2 ) of C as λ variates continuously. In particular,
we have limλ→√−c rλ,j = ∞.
Now we define complex focal radii for a Fredholm submanifold in a pseudo be a Fredholm submanifold in a pseudo-Hilbert space V .
Hilbert space. Let M
and X ∈ Tx M.
Denote
the shape tensor of M.
Let v ∈ Tx⊥ M
Denote by A
by γv the geodesic in V with γ̇v (0) = v. The Jacobi field J along γv with
v X is given by J (s) = X − s A
v X, where we identify
J (0) = X and J (0) = −A
along γv is catched as a real number s0 with
Tγv (s)V with V . A focal radius of M
v ) = {0}, where id is the identity transformation of Tx M.
So, we call a
Ker(id −s0 A
c
c
along γv
complex number z0 with Ker(id −z0 Av ) = {0} a complex focal radius of M
c
c
c
and call dimc Ker(id −z0 Av ) the multiplicity of the complex focal radius z0 , where A
c
(respectively id ) is the complexification of A (respectively id). Also, for a complex
along γv , we call z0 v (∈ T ⊥ M c ) a complex focal normal of M
focal radius z0 of M
x
at x. Furthermore, we call the set of all complex focal normals of M at x a complex
at x.
focal set of M
Now we define the complex equifocality of a submanifold in a symmetric space of
non-compact type. Let M be an immersed submanifold with globally flat and abelian
180
N. Koike
normal bundle in a symmetric space N = G/K of non-compact type. Let ṽ be a
parallel unit normal vector field of M. Assume that the number (which may be zero
and infinite) of distinct complex focal radii along γṽx is independent of the choice of
x ∈ M. Let {rix | i = 1, 2, . . . } be the set of all complex focal radii along γṽx , where
‘|ri,x | < |ri+1,x |’ or ‘|ri,x | = |ri+1,x | and Re(ri,x ) > Re(ri+1,x )’ or ‘|ri,x | = |ri+1,x |
and Re(ri,x ) = Re(ri+1,x ) and Im(ri,x ) = −Im(ri+1,x ) < 0’. Let ri (i = 1, 2, . . . )
be complex-valued functions on M defined by assigning ri,x to each x ∈ M. We call
these functions ri (i = 1, 2, . . . ) complex focal radius functions for ṽ. We call ri ṽ a
complex focal normal vector field for ṽ. If, for each parallel unit normal vector field
ṽ of M, the number of distinct complex focal radii of ṽx is independent of the choice
of x ∈ M and, furthermore, each complex focal radius function for ṽ is constant on
M (in the case where the number is not equal to zero), then we call M a complex
equifocal submanifold.
4. The parallel transport map for a semi-simple non-compact Lie group
Let G be a semi-simple non-compact Lie group and g be its Lie algebra. Take an
Ad(G)-invariant non-degenerate symmetric bilinear form ·, · of g. Denote by the
same symbol ·, · the bi-invariant metric of G induced from ·, ·. Fix the orthogonal
time-space decomposition g = g− ⊕ g+ . Set ·, ·g± := −πg∗− ·, · + πg∗+ ·, ·,
where πg− (respectively πg+ ) is the projection of g onto g− (respectively g+ ). Let
H 0 ([0, 1], g) be the space of all L2 -integrable paths u : [0, 1] → g (with respect
to ·, ·g± ). Note that H 0 ([0, 1], g) is independent of the choice of the orthogonal
time-space decomposition g = g− ⊕ g+ .
Let H 0 ([0, 1], g− ) (respectively H 0 ([0, 1], g+ )) be the space of all L2 -integrable
paths u : [0, 1] → g− (respectively u : [0, 1] → g+ ) with respect to −·, ·|g− ×g−
(respectively ·, ·|g+ ×g+ ). It is clear that H 0 ([0, 1], g) = H 0 ([0, 1], g− ) ⊕
Define a non-degenerate
symmetric bilinear form ·, ·0 of
H 0 ([0, 1], g+ ).
1
0
It is easy to show that
H ([0, 1], g) by u, v0 :=
0 u(t), v(t) dt.
the decomposition H 0 ([0, 1], g) = H 0 ([0, 1], g− ) ⊕ H 0 ([0, 1], g+ ) is an
orthogonal time-space decomposition with respect to ·, ·0 . For simplicity, set
H±0 := H 0 ([0, 1], g± ) and ·, ·0,H 0 := −π ∗ 0 ·, ·0 + π ∗ 0 ·, ·0 , where πH 0
±
H−
H+
−
(respectively πH 0 ) is the projection
of H 0 ([0, 1], g) onto H−0 (respectively H+0 ).
1
+
Hence
It is clear that u, v0,H 0 = 0 u(t), v(t)g± dt (u, v ∈ H 0 ([0, 1], g)).
±
(H 0 ([0, 1], g), ·, ·0,H 0 ) is a Hilbert space, that is, (H 0 ([0, 1], g), ·, ·0 ) is a pseudo±
Hilbert space. Let H 1 ([0, 1], G) be the Hilbert Lie group of all absolutely continuous
paths g : [0, 1] → G such that g is squared integrable (with respect to ·, ·g± ), that
Submanifold geometries in a symmetric space
181
is, g∗−1 g ∈ H 0 ([0, 1], g). Define a map φ : H 0 ([0, 1], g) → G by φ(u) = gu (1)
(u ∈ H 0 ([0, 1], g)), where gu is the element of H 1 ([0, 1], G) satisfying gu (0) = e
−1 g = u. We call this map the parallel transport map (from 0 to 1).
and gu∗
u
Set P (G, e × G) := {g ∈ H 1 ([0, 1], G) | g(0) = e} and e (G) := {g ∈
H 1 ([0, 1], G) | g(0) = g(1) = e}. Here we note that e (G) is not necessarily
connected because we do not assume the simply connectedness of G. The group
H 1 ([0, 1], G) acts on H 0 ([0, 1], g) by gauge transformations, that is,
g ∗ u := Ad(g)u − g g∗−1
(g ∈ H 1 ([0, 1], G),
u ∈ H 0 ([0, 1], g)).
It can be shown that the following facts hold:
(i) the above action of H 1 ([0, 1], G) on H 0 ([0, 1], g) is isometric;
(ii) the above action of P (G, e × G) on H 0 ([0, 1], g) is transitive and free;
(iii) φ(g ∗ u) = g(0)φ(u)g(1)−1 for g ∈ H 1 ([0, 1], G) and u ∈ H 0 ([0, 1], g);
(iv) φ : H 0 ([0, 1], g) → G is regarded as a e (G)-bundle;
(v) if φ(u) = x0 φ(v)x1−1 (u, v ∈ H 0 ([0, 1], g), x0 , x1 ∈ G), then there exists
g ∈ H 1 ([0, 1], G) such that g(0) = x0 , g(1) = x1 and u = g ∗ v. In particular,
it follows that each u ∈ H 0 ([0, 1], g) is described as u = g ∗ 0̂ in terms of some
g ∈ P (G, e × G), where 0̂ is the constant path at the zero element 0 of g.
Imitating the proof of Lemma 2.1 in [Ko3], we can show the following relation.
1
L EMMA 4.1. For v ∈ Tu H 0 ([0, 1], g), φ∗u (v) = ( 0 Ad(g −1 )v dt)g(1)−1
∗ , where
0
0
u = g ∗ 0̂ (g ∈ P (G, e × G)) and we identify Tu H ([0, 1], g) with H ([0, 1], g).
Let ξ ∈ H 0 ([0, 1], g) with ξ(0) = 0 and s → gs be a curve in P (G, e × G)
s
0
0
with g0 = ê and dg
ds |s=0 = ξ . For u ∈ H ([0, 1], g), define ξ(u) ∈ Tu H ([0, 1], g)
d
by ξ(u) := ds (gs ∗ u)|s=0 . Imitating the proof of Lemma 2.2 in [Ko3], we have the
following relation.
L EMMA 4.2. For ξ ∈ H 0 ([0, 1], g) with ξ(0) = 0 and u = g ∗ 0̂ ∈ H 0 ([0, 1], g), the
relation ξ(u) = −[ξ, g g∗−1 ] − ξ holds.
the Levi–Civita connection of (G, ·, ·)
Denote by ∇ ∗ (respectively ∇)
0
(respectively (H ([0, 1], g), ·, ·0 )). Imitating the proof of Lemma 2.3 in [Ko3], we
can obtain the following relation.
L EMMA 4.3. For the horizontal lift v L of a vector field v on G to H 0 ([0, 1], g) and
ξ(u) ∈ Tu H 0 ([0, 1], g), we have
L
L
ξ(u) v L = (∇ ∗
∇
φ∗ ξ(u) v)u + [ξ, (vφ(u) )u ]
−1
−1 L
+ 12 (φ(u)∗ [φ(u)−1
∗ (φ∗ (ξ(u))), Ad(φ(u) )(vφ(u) φ(u)∗ )])u .
182
N. Koike
In terms of Lemma 4.1, we obtain the following fact.
P ROPOSITION 4.4. The submersion φ : (H 0 ([0, 1], g), ·, ·0 ) → (G, ·, ·) is a
pseudo-Riemannian submersion.
Proof. Let (e1 , . . . , em− ) (respectively (ē1 , . . . , ēm+ )) be an
√ orthonormal basesi of g−
(respectively g+ ) with respect to ·, ·. Set lecoi ,k :=
2ei cos(2kπt), lei ,k :=
√
√
√
co
si
2ei sin(2kπt), lēj ,k :=
2ēj cos(2kπt) and lēj ,k :=
2ēj sin(2kπt) (i =
1, . . . , m− , k ∈ N). Also, denote by êi (respectively ēˆj ) the constant path at
ei (respectively ēj ). It can be shown that lecoi ,k , lesii ,k (i = 1, . . . , m− , k ∈ N)
and êi (i = 1, . . . , m− ) construct an orthonormal base of (H−0 , ·, ·0 |g− ×g− ) and
that lēcoj ,k , lēsij ,k (j = 1, . . . , m+ , k ∈ N) and ēˆj (j = 1, . . . , m+ ) construct that
of (H+0 , ·, ·0 |g+ ×g+ ). The vectors lecoi ,k , lesii ,k (i = 1, . . . , m− , k ∈ N) and êi
(i = 1, . . . , m− ) are unit time-like vectors with respect to ·, ·0 . Also, the vectors
lēcoj ,k , lēsij ,k (j = 1, . . . , m+ , k ∈ N) and ēˆj (j = 1, . . . , m+ ) are unit spacelike
vectors with respect to ·, ·0 . We identify T0̂ H 0 ([0, 1], g) with H 0 ([0, 1], g).
From Lemma 4.1, we have φ∗0̂ (lecoi ,k ) = φ∗0̂ (lesii ,k ) = φ∗0̂ (lēcoj ,k ) = φ∗0̂ (lēsij ,k ) = 0
(i = 1, . . . , m− , j = 1, . . . , m+ , k ∈ N), φ∗0̂ (êi ) = ei (i = 1, . . . , m− ) and
φ∗0̂ (ēˆj ) = ēj (j = 1, . . . , m+ ). Also, it is easy to show that êi (i = 1, . . . , m− ) and
ēˆj (j = 1, . . . , m+ ) are orthogonal to lecoi ,k , lesii ,k , lēcoj ,k and lēsij ,k (i = 1, . . . , m− ,
j = 1, . . . , m+ , k ∈ N) with respect to ·, ·0 . Set T− := Span{lecoi ,k | i =
1, . . . , m− , k ∈ N} ⊕ Span{lesii ,k | i = 1, . . . , m− , k ∈ N} and T+ := Span{lēcoi ,k | i =
1, . . . , m+ , k ∈ N} ⊕ Span{lēsii ,k | i = 1, . . . , m+ , k ∈ N}, which give the orthogonal
time-space decomposition of T0̂ φ −1 (e). Denote by ι the inclusion map of φ −1 (e) into
H 0 ([0, 1], g). The above facts imply that (T0̂ φ −1 (e), ι∗ ·, ·0,T± ) is a Hilbert space
and that the restriction φ∗0̂ |T ⊥ φ −1 (e) of φ∗0̂ to T ⊥ φ −1 (e) is a linear isometry onto
0̂
0̂
(Te G, ·, ·), where T ⊥ φ −1 (e) is the normal space of φ −1 (e) at 0̂ with respect to ·, ·0 .
0̂
Take an arbitrary u ∈ H 0 ([0, 1], g). Let u = g ∗ 0̂ (g ∈ P (G, e × G)). In terms of
Lemma 4.1, we can show that Ad(g)T0̂ φ −1 (e) = Tu φ −1 (φ(u)), Ad(g)T ⊥ φ −1 (e) =
0̂
−1
Tu⊥ φ −1 (φ(u)) and φ∗u ◦ Ad(g) = Rg(1)∗
◦ φ∗0̂, where we identify T0̂ H 0 ([0, 1], g) and
Tu H 0 ([0, 1], g) with H 0 ([0, 1], g). Also, the operator Ad(g) is a linear isometry of
(Hg0± ([0, 1], g), ·, ·0 ). These facts deduce that (Tu φ −1 (φ(u)), ι∗u ·, ·0,Ad(g)T± ) is
also a Hilbert space and that φ∗u |Tu⊥ φ −1 (φ(u)) is also a linear isometry, where ιu is the
inclusion map of φ −1 (φ(u)) into H 0 ([0, 1], g). Therefore, from the arbitrariness of u,
we can recognize that φ is a pseudo-Riemannian submersion.
2
Submanifold geometries in a symmetric space
183
5. The spectrum of the shape operators of the lifted submanifolds
Let N = G/K be a symmetric space of non-compact type and π be the natural
projection of G onto G/K. Let (g, σ ) be the orthogonal symmetric Lie algebra of
G/K, f = {X ∈ g | σ (X) = X} and p = {X ∈ g | σ (X) = −X}, which is identified
with the tangent space TeK N. Let ·, · be the Ad(G)-invariant non-degenerate
symmetric bilinear form of g inducing the Riemannian metric of N. Note that ·, ·|f×f
(respectively ·, ·|p×p ) is negative (respectively positive) definite. Denote by the same
symbol ·, · the bi-invariant pseudo-Riemannian metric of G induced from ·, · and
the Riemannian metric of N. Set g+ := p and g− := f. Let (H 0 ([0, 1], g), ·, ·0 )
be a pseudo-Hilbert space as stated in Section 4 and φ : H 0 ([0, 1], g) → G be the
the Levi–Civita
:= π ◦ φ. Denote by ∇, ∇ ∗ and ∇
parallel transport map. Set φ
0
connection of N, (G, ·, ·) and (H ([0, 1], g), ·, ·0 ), respectively. Let M be an
be that
immersed submanifold in N. Let M ∗ be one component of π −1 (M) and M
−1
∗
∗
of φ (M ). It is clear that M (respectively M) is an immersed submanifold in G
the shape tensors of M, M ∗ and
(respectively H 0 ([0, 1], g)). Denote by A, A∗ and A
0
respectively. As stated in Section 4, (H ([0, 1], g), ·, · 0 ) is a Hilbert space,
M,
0,H±
where H±0 := H 0 ([0, 1], g± ). Denote by the same symbol ·, ·0,H 0 the Riemannian
±
induced from ·, · 0 . In this section, we investigate the spectrum of
metric of M
0,H±
First we show the following fact.
the shape operator of M.
is a pseudo-Riemannian Hilbert submanifold
P ROPOSITION 5.1. The submanifold M
0
in (H ([0, 1], g), ·, ·0 ).
into H 0 ([0, 1], g). We have only to show
Proof. Denote by ι the inclusion map of M
∗
ι ·, ·0 ) is a pseudo-Riemannian Hilbert manifold. Fix an arbitrary u0 ∈ M.
that (M,
For each u ∈ U , take
Let U be a sufficiently small neighborhood of u0 in M.
g u ∈ H 1 ([0, 1], G) such that g u ∗ u = 0̂ and g u (1) = e. The existence of such a
g u is shown in terms of the facts (i)–(v) stated in Section 4. Furthermore, we can take
g u , as the correspondence u → g u is a smooth map of U into H 1 ([0, 1], G). Since
contains Ker φ
(and hence H−0 ).
φ, the space (g u ∗ ·)∗ (Tu M)
φ ◦ (g u ∗ ·) = g u (0) ◦ ∗0̂
Hence we have
0
0
= ((g u ∗·)∗ (Tu M)∩H
(g u ∗·)∗ (Tu M)
+ )⊕H−
(⊂ T0̂ H 0 ([0, 1], g) = H 0 ([0, 1], g)).
0
u
0
u
u
−1
Set W+u := (g u ∗ ·)−1
∗ ((g ∗ ·)∗ (Tu M) ∩ H+ ) and W− := (g ∗ ·)∗ (H− ). It is clear
u
u
u
= W− ⊕ W+ (orthogonal timethat u → W± are distributions on U such that Tu M
·, ·0,W u ) (u ∈ U ) are
space decomposition). Furthermore, it is shown that (Tu M,
±
ι∗ ·, ·0 )
Hilbert spaces. Therefore, it follows from the arbitrariness of u0 that (M,
184
N. Koike
is a pseudo-Riemannian Hilbert
is a pseudo-Riemannian Hilbert manifold, that is, M
0
2
submanifold in (H ([0, 1], g), ·, ·0 ).
Also, we have the following fact.
be as above. Take u ∈ M.
Set g0 := φ(u)
P ROPOSITION 5.2. Let M, M ∗ and M
1
and x := φ(u). Then there exists g ∈ H ([0, 1], G) satisfying g ∗ u = 0̂, g(0) = g0−1
= g(0)M ∗ (and hence = g(0)M).
(and hence g(0)x = eK) and φ(g ∗ M)
φ(g ∗ M)
Proof. According to the facts (iii) and (iv) stated in Section 4, we can find g ∈
H 1 ([0, 1], G) satisfying g(0) = g0−1 , g(1) = e and g ∗ u = 0̂. Then it follows
= g −1 M ∗ .
2
from fact (iii) that φ(g ∗ M)
0
Denote by X∗ (respectively XL ) the horizontal lift of a vector (or a vector field) X
of N to G (respectively of G to H 0 ([0, 1], g)). In particular, the horizontal lift Xe∗ of
φ −1 (g0 K),
X ∈ TeK N to e is identified with X. Fix v ∈ Tg⊥0 K M with v = 1 and u ∈ where g0 ∈ M ∗ . According to Proposition 5.2, we may assume g0 K = eK and u = 0̂.
Also, we may assume e ∈ M ∗ . Take a maximal abelian subspace h (in p) containing v.
Let p = h + α∈+ pα be the root space decomposition. Take a maximal abelian
subalgebra h̃ in g containing h and let hf := h̃ ∩ f. Denote by X̂ the constant path at
X ∈ g. Note that X̂ coincides with the horizontal lift of X to 0̂ under the identification
of T0̂ H 0 ([0, 1], g) and H 0 ([0, 1], g) (see the proof of Proposition 4.4).
For X ∈ pα , let Xf be the element of f such that ad(a)X = α(a)Xf for all a ∈ h.
Note that ad(a)Xf = α(a)X for all a ∈ h. For X ∈ pα (respectively h̃), we define loop
i
∈ H 0 ([0, 1], g)C (i = 1, 2, j ∈ Z) by
vectors lX,j
1
lX,j
(t) := X cos(2j πt) +
√
−1Xf sin(2j πt)
1
(respectively lX,j
(t) := X cos(2j πt))
and
2
(t) :=
lX,j
√
−1X sin(2j πt) + Xf cos(2j πt)
√
2
(t) := −1X sin(2j πt)).
(respectively lX,j
Note that these loop vectors are vertical with respect to φ (see the proof of
Proposition 4.4).
Under the identification of H 0 ([0, 1], g)C and T0̂ H 0 ([0, 1], g)C , the constant path
i
(i = 1, 2) are regarded as elements of T0̂ H 0 ([0, 1], g)C.
X̂ and loop vectors lX,j
Now we prepare the following lemma.
Submanifold geometries in a symmetric space
185
L EMMA 5.3. Let {X1 , . . . , Xn } be an orthonormal base of TeK M, {Y1 , . . . , Ymf } be
0 } be that of h̃ and {e α , . . . , e α } (α ∈ ) be that of p . Then the
that of f, {e10, . . . , em
+
α
mα
1
0
system
{X̂1 , . . . , X̂n } ∪ {Ŷ1 , . . . , Ŷmf }
√
∪ { 2lei 0 ,k | i = 1, 2, j = 1, . . . , m0 , k ∈ N}
j
i
∪
{leα ,k | i = 1, 2, j = 1, . . . , mα , k ∈ Z \ {0}}
α∈+
j
c , ·, ·H ), where ·, ·H is the non-degenerate
is a pseudo-orthonormal base of (T0̂ M
0
0
c
Hermitian bilinear form of T0̂ M defined from ·, ·0 .
c = (T M
∩ T ⊥ φ −1 (e))c ⊕ T φ −1 (e)c (orthogonal direct
Proof. Clearly we have T0̂ M
0̂
0̂
0̂
sum). It is easy to show that {X̂1 , . . . , X̂n } ∪ {Ŷ1 , . . . , Ŷmf } is an orthonormal base of
∩ T ⊥ φ −1 (e))c and that the closure of the orthogonal direct sum
(T0̂ M
0̂
√
Span{ 2lei 0 ,k | i = 1, 2, j = 1, . . . , m0 , k ∈ N}
j
⊕
Span{lei α ,k | i = 1, 2, j = 1, . . . , mα , k ∈ Z \ {0}}
α∈+
j
√
coincides with T0̂ φ −1 (e)c . Also, by simple calculations, it can be shown that { 2l i 0
i = 1, 2, j = 1, . . . , m0 } is an orthonormal system and that
|lei1α ,k1 ,
j
1
|
ej ,k
i2
H
leα ,k2 0 | =
j
2
δi1 i2 δj1 j2 δk1 ,−k2 (1 ≤ i1 , i2 ≤ 2, 1 ≤ j1 , j2 ≤ mα , k1 , k2 ∈ Z \ {0}), where δ·· is the
Kronecker symbol. Thus we obtain the statement.
2
From Lemmas 4.2 and 4.3, the following relations are directly deduced.
L EMMA 5.4. Let ṽ be a vector field on a neighborhood of e in G with ṽe = v
(∈ TeK N ∼ (TeK N)L
e ⊂ Te G). Then the following statements hold.
ṽ L = (∇ ∗ ṽ)L holds.
(i) For X ∈ h̃, the relation ∇
X 0̂
X̂
ṽ L = (∇ ∗ ṽ)L + tα(v)Xf − 1 α(v)X̂f and
(ii) For X ∈ pα , the relations ∇
X 0̂
X̂
2
1
L
∗
L
∇X̂f ṽ = (∇Xf ṽ) + tα(v)X − 2 α(v)X̂ hold.
0̂
ci ṽ L = −[ t l i (t) dt, v] (i = 1, 2, j ∈ Z) hold,
(iii) For X ∈ g, the relations ∇
0 X,j
lX,j
c is the complexification of ∇.
where ∇
Remark 5.1. If N is of compact type, then the α(v) in the relations of the statement
(ii) are replaced by −α(v).
186
N. Koike
Also, since π is a pseudo-Riemannian submersion, we have the following
relations.
L EMMA 5.5.
1
∗
∗
(i) For X ∈ TeK M (∼(TeK M)L
e ⊂ Te M ), the relation Av X = Av X + 2 [v, X]
holds.
∗ v ∗ = 1 [v, X] holds.
(ii) For X ∈ f (⊂ Te G), the relation ∇X
2
Proof. Since π is a pseudo-Riemannian submersion, the statement (i) is shown by
imitating the proof of Proposition 7.3 in [TT]. Also, since v ∗ is a right-invariant
vector field along the fiber K of π, the statement (ii) is directly deduced.
2
From Lemmas 5.4 and 5.5, the following facts are directly deduced.
P ROPOSITION 5.6. Let X ∈ h̃. Then the following statements hold.
v̂ X̂ = λX̂.
(i) If X ∈ h ∩ TeK M and Av X = λX, then we have A
v̂ X̂ = 0.
(ii) If X ∈ hf , then we have A
i
c l i = 0.
(iii) For the vertical loop vectors lX,j
(i = 1, 2, j ∈ N), we have A
v̂ X,j
In the following, for a family {ak }k∈Z\{0} of complex numbers or vectors, the
notation k∈Z\{0} ak implies ∞
k=1 (ak + a−k ), where Z is the set of all integers.
Also, we obtain the following relations in terms of Lemmas 5.4 and 5.5.
P ROPOSITION 5.7. Let X ∈ pα . The following statements hold.
v̂ X̂f =
v̂ X̂ = λX̂, A
(i) If X ∈ TeK M, Av X = λX and α(v) = 0, then we have A
c l i = 0 (i = 1, 2, j ∈ Z).
A
v̂ X,j
(ii) If X ∈ TeK M, Av X = λX (λ = ±α(v)) and α(v) = 0, then we have
√
bλ,α
−1
α(v)
1
2
c
−
(l + aλ,α lX,k ) ∈ Ker A
X̂ + aλ,α X̂f +
id ,
v̂
b − kπ X,k
2bλ,α
k∈Z\{0} λ,α
√
where aλ,α = α(v)/(λ + λ2 − α(v)2 ) (two-valued), bλ,α = arctan(aλ,α −1)
c .
(∞-valued) and id is the identity transformation of T0̂ M
(iii) If X ∈ TeK M, Av X = ±α(v)X and α(v) = 0, then there exists no eigenvector
1
2
c belonging to Span({X̂, X̂f } ∪ (
of A
k∈Z\{0} {lX,k , lX,k })).
v̂
⊥ M and α(v) = 0, then we have A
v̂ X̂f = A
c l i = 0 (i = 1, 2, k ∈
(iv) If X ∈ TeK
v̂ X,k
Z \ {0}).
√
⊥ M and α(v) = 0, then we have A
c l 1 = (α(v) −1/(2j π))l 1
(v) If X ∈ TeK
X,j
v̂ X,j
Submanifold geometries in a symmetric space
187
(j ∈ Z \ {0}) and
√
α(v) −1
2j + 1
2
c
l
id
∈ Ker Av̂ −
X̂f +
2j − 2k + 1 X,k
(2j + 1)π
k∈Z\{0}
(j ∈ Z).
Remark 5.2.
(i) The (∞-valued) quantity bλ,α in statement (ii) is precisely described as

√

jπ
α(v)
−1


+
arctanh
(j ∈ Z)
(|λ| > |α(v)|),

2
2
λ
bλ,α =
√

2j + 1
λ
−1


π+
arctanh
(j ∈ Z) (|λ| < |α(v)|).

4
2
α(v)
(ii)
According to statement (iii), in the case where M is a horosphere in a hyperbolic
c . Hence the spectrum
space, there exists no point spectre (i.e. eigenvalue) of A
v̂
c
consists of only zero, which is a continuous spectre.
of A
v̂
c by A
v̂ for simplicity. Let ṽ be a
Proof of Proposition 5.7. In this proof, denote A
v̂
vector field on a neighborhood of e in G with ṽe = v (∈ TeK N = (TeK N)L
e ⊂ Te G).
Assume that X ∈ pα ∩ TeK M and Av X = λX. Then it follows from (ii) of Lemma 5.4
that
1
1
ṽ L = (∇ ∗ ṽ)L − α(v)
lX,k
,
√
∇
X 0̂
X̂
2π −1 k∈Z\{0} k
that is,
v̂ X̂ = (A∗v X)L +
A
0̂
1
α(v)
1
lX,k
.
√
2π −1 k∈Z\{0} k
On the other hand, it follows from (i) of Lemma 5.5 that A∗v X = λX + (α(v)/2)Xf .
Hence we have
1
1
v̂ X̂ = λX̂ + α(v) X̂f + α(v)
lX,k
.
(5.1)
√
A
2
2π −1 k∈Z\{0} k
Similarly, it follows from (ii) of Lemma 5.4 and (ii) of Lemma 5.5 that
1
2
v̂ X̂f = − α(v) X̂ + α(v)
lX,k
A
.
√
2
2π −1 k∈Z\{0} k
Also, it follows from (iii) of Lemma 5.4 that
√
√
α(v) −1 1
α(v) −1 2
1
2
(lX,k − X̂), Av̂ lX,k =
(lX,k − X̂f ).
Av̂ lX,k =
2kπ
2kπ
(5.2)
(5.3)
188
N. Koike
From (5.1)–(5.3), statement (i) is directly deduced. Assume α(v) = 0. We solve the
equation
1
2
(x1,k lX,k + x2,k lX,k ) = 0.
(Av̂ − x1 id) x2 X̂ + x3 X̂f +
k∈Z\{0}
From (5.1)–(5.3), the following relations are deduced:
√
α(v) −1 x1,k
= 0,
2(λ − x1 )x2 − α(v)x3 −
π
k
k∈Z\{0}
√
α(v) −1 x2,k
= 0,
α(v)x2 − 2x1x3 −
π
k
k∈Z\{0}
√
√
(α(v) −1 − 2kπx1 )x1,k = α(v) −1x2 ,
√
√
(α(v) −1 − 2kπx1 )x2,k = α(v) −1x3 .
(5.4)
If x1 = 0 or x2 = 0,then these relations deduce that x3 = xik = 0 (i = 1, 2, k ∈
Z \ {0}). So, we may assume xi = 0 (i = 1, 2). Furthermore, assume x2 = 1. Then it
follows from (5.4) that
√
√
α(v) −1
α(v)x3 −1
, x2k =
,
x1k =
√
√
α(v) −1 − 2kπx1
α(v) −1 − 2kπx1
∞
4α(v)2 x1
2(λ − x1 ) − α(v)x3 −
= 0,
(5.5)
2
2 2 2
k=1 α(v) + 4k π x1
α(v) − 2x1 x3 −
∞
k=1
4α(v)2 x1 x3
= 0.
α(v)2 + 4k 2 π 2 x12
Substituting t = 0 into the Fourier expansion
cosh(θ (t − 1)) =
∞
2θ sinh θ
sinh θ +
cos(kπt)
2 + k2π 2
θ
θ
k=1
of cosh(θ (t − 1)) (∈ L2 (0, 2)), we have
coth θ =
1
1
+
√ .
θ k∈Z\{0} θ + kπ −1
In general,√
this relation holds for any complex number θ . From this relation, coth z =
√
−1/tan( −1z) and (5.5), we have
√
α(v) −1
bλ,α
bλ,α aλα
and x2,k =
,
, x1,k =
x3 = aλ,α , x1 =
2bλ,α
bλ,α − kπ
bλ,α − kπ
Submanifold geometries in a symmetric space
189
where
√ aλ,α and bλ,α are as in statement (ii). However, if λ = ±α(v), then we have
α(v) −1/2bλ,α = 0, which contradicts x1 = 0. Therefore, we obtain statements (ii)
⊥ M. Then it follows
and (iii). Next we show statements (iv) and (v). Let X ∈ pα ∩ TeK
from (ii) of Lemma 5.4 and (ii) of Lemma 5.5 that
v̂ X̂f =
A
1
α(v)
2
lX,k
.
√
2π −1 k∈Z\{0} k
Also, it follows from (iii) of Lemma 5.4 that
√
√
v̂ l 2 = α(v) −1 (l 2 − X̂f ).
v̂ l 1 = α(v) −1 l 1 , A
A
X,k
X,k
X,k
X,k
2kπ
2kπ
(5.6)
(5.7)
From (5.6) and (5.7), statement (iv) and the first fact of statement (v) are directly
2 |k ∈
deduced. Assume α(v) = 0. From (5.6) and (5.7), we see that Span({X̂f } ∪ {lX,k
v̂ -invariant. Now we find eigenvectors belonging to Span({X̂f } ∪ {l 2 |
Z \ {0}}) is A
X,k
2 ) = 0.
v̂ − x1 id)(x2 X̂f +
x
l
k ∈ Z \ {0}}). Solve the equation (A
2,k
k∈Z\{0}
X,k
2
| k ∈ Z \ {0}} by (5.7),
Since there exists no eigenvector belonging to Span{lX,k
we may assume x2 = 1. From (5.6) and the second relation of (5.7), the following
relations are deduced:
√
α(v) −1 1
α(v)
x2,k , x2,k =
x1 = −
.
(5.8)
√
2π
k
α(v)
+
2kπ −1x1
k∈Z\{0}
From these relations and
coth θ =
1
1
+
√ ,
θ k∈Z\{0} θ + kπ −1
√
we have coth[α(v)/(2x1 )] = 0, that is, x1 = (α(v) −1)/((2j + 1)π) (j ∈ Z). Then
we have x2,k = (2j + 1)/(2j − 2k + 1). Thus the second fact of statement (v) is
deduced.
2
In terms of Proposition 5.1, Lemma 5.3, Proposition 5.6 and the proof of
Proposition 5.7, we obtain the following fact.
is a Fredholm submanifold in
T HEOREM 5.8. The lifted submanifold M
0
(H ([0, 1], g), ·, ·0 ).
is a pseudo-Riemannian Hilbert submanifold.
Proof. According to Proposition 5.1, M
Also, according to Lemma 5.3, Proposition 5.6 and the proof of Proposition 5.7, each
190
N. Koike
is a compact operator with respect to ι∗ ·, ·H
complexified shape operator of M
into
where ι is the inclusion map of M
H 0 ([0, 1], g)
and
·, ·H 0
0,H±
0,H±0
,
is the Hermitian
is a compact
inner product defined from ·, ·0,H 0 . Hence, each shape operator of M
±
is a Fredholm submanifold.
2
operator with respect to ι∗ ·, · 0 , that is, M
0,H±
c implies
In the following, the spectrum of the complexified shape operator A
v̂
c
c
∗
H
c
∗
H
that of Av̂ : (T M , ι ·, ·0,H± ) → (T M , ι ·, ·0,H± ). In terms of Lemma 5.3,
Propositions 5.6 and 5.7, we obtain the following result in the case where M is a
curvature adapted submanifold in a symmetric space of non-compact type.
T HEOREM 5.9. Let M be an immersed curvature adapted submanifold in a symmetric
: H 0 ([0, 1], g) → N be the above pseudospace N = G/K of non-compact type, φ
Riemannian submersion and M be one component of φ −1 (M). Fix a unit normal
where we may assume that the base point of ṽ is 0̂ (see Proposition 5.2).
vector ṽ of M,
Set v := φ∗ (ṽ). Let p = h + α∈+ pα be the root space decomposition for a fixed
maximal abelian subspace h containing v and set v := {α ∈ + | α(v) = 0} and
⊥ M = {0}}. Denote by A (respectively A
c ) the shape
⊥ := {α ∈ + | pα ∩ TeK
If the spectrum of Av
tensor of M (respectively the complexified shape tensor of M).
c is given by
is {λ1 , . . . , λg }, then the spectrum of A
ṽ
{0} ∪ λi i ∈
Iα
α∈v ∪{0}
+
∪
√
i ∈ Iα , j ∈ Z
α∈+ \v arctanh[α(v)/λi ] + j π −1
α(v)
−
∪
√
i ∈ Iα , j ∈ Z
1
α∈+ \v arctanh[λi /α(v)] + (j + 2 )π −1
√
α(v) −1 j
∈
Z
\
{0}
,
(5.9)
∪
jπ
α∈ \
⊥
α(v)
v
where Iα := {i | pα ∩ Ker(Av − λi id) = {0}} (α ∈ + ∪ {0}), Iα+ := {i ∈ Iα |
|λi | > |α(v)|} (α ∈ + ) and Iα− := {i ∈ Iα | |λi | < |α(v)|} (α ∈ + ). Furthermore,
c
if the sum
in (5.9) are the direct sum, then the multiplicities of eigenvalues of A
ṽ
are given as Table 1. Also, if the ±α(v) (α ∈ + \ v ) are not principal curvatures
c consisting of the
of direction v, then there exists a pseudo-orthonormal base of T0̂ M
c .
eigenvectors of A
ṽ
Submanifold geometries in a symmetric space
191
TABLE 1.
Eigenvalue
Multiplicity
0
λi
∞
dim(pα ∩ Ker(Av − λi id))
α∈v ∪{0}
α(v)
√
arctanh[α(v)/λi ] + j π −1
dim(pα ∩ Ker(Av − λi id))
α(v)
dim(pα ∩ Ker(Av − λi id))
√
arctanh[λi /α(v)] + (j + 12 )π −1
√
α(v) −1
2j π
√
α(v) −1
(2j + 1)π
⊥ M)
dim(pα ∩ TeK
⊥ M)
dim(pα ∩ TeK
Proof. Since M is curvature adapted, we have
TeK M =
g
(pα ∩Ker(Av −λi id))
i=1 α∈+ ∪{0}
⊥
and TeK
M=
α∈+ ∪{0}
⊥
(pα ∩TeK
M).
These facts together with Lemma 5.3, Propositions 5.6, 5.7 and Remark 5.2 deduce
the statement of this theorem.
2
Remark 5.3. Consider the case where M is the principal orbit through exp Y (Y ∈ h)
of the isotropy representation of N = G/K. We may assume exp Y = eK by
⊥ M. Let p = h +
operating g −1 on M, where exp Y = gK. Take v ∈ TeK
α∈+ pα
⊥ M. Then the spectrum of A is
be the root space decomposition, where h = TeK
v
given by {α(v)/tanh α(Y ) | α ∈ + } and Ker(Av − (α(v)/tanh α(Y )) id) = pα if
α(v)/tanh α(Y ) (α ∈ + ) are mutually distinct. Hence, according to Theorem 5.9,
c is given by
the spectrum of A
v̂
α(v)
α ∈ + \ v , j ∈ Z
{0} ∪
√
α(Y ) + j π −1 c = ∞. Furthermore, if
and dimc Ker A
v̂
α(v)
√
α(Y ) + j π −1
(α ∈ + \ v , j ∈ Z)
192
N. Koike
are mutually distinct, then we have
α(v)
c −
√
id
= dim pα
dimc Ker A
v̂
α(Y ) + j π −1
(α ∈ + \ v , j ∈ Z).
From Theorem 5.9, we obtain the following result for a curvature adapted
hypersurfaces in a rank-one symmetric space of non-compact type.
C OROLLARY 5.10. Let M be an immersed curvature adapted hypersurface in a
rank-one symmetric space N of non-compact type other than the hyperbolic space
the number of whose principal curvatures is g on M, and let {λ1 , . . . , λg } be its
principal curvature functions, where a unit normal vector field v of M is fixed.
be as in Theorem 5.9, let c be the minimal sectional curvature
Also, let φ, A and A
of N and let D be the eigendistribution of R(·, v)v|T M for c. Assume that |λi |
√
√
(i = 1, . . . , g) are distinct from −c and −c/2 at each point of M. Set
√
ID± := {i | Ker(Av − λi id) ∩ D = {0}, ±(|λi | − −c) > 0} and ID±⊥ := {i |
√
Ker(Av − λi id) ∩ D ⊥ = {0}, ±(|λi | − −c/2) > 0}. Then the set of all complex
of principal curvature functions of one component M
φ −1 (M) is given by
√
−c
+
i
∈
I
,
j
∈
Z
{0} ∪
√
√
D
arctanh[ −c/λi ◦ φ] + j π −1 √
−c
−
i∈I , j ∈Z
∪
√
√
D
arctanh[λi ◦ φ/ −c] + (j + 12 )π −1 √
−c
i ∈ I +⊥ , j ∈ Z
∪
√
√
D
2 arctanh[ −c/(2(λi ◦ φ))] + 2j π −1 √
−c
−
i ∈ ID ⊥ , j ∈ Z .
∪
√
√
2 arctanh[2(λ ◦ φ)/ −c] + (2j + 1)π −1 i
Furthermore, if, for all (i1 , i2 ) ∈ ID+ × ID+⊥ (respectively ID− × ID−⊥ ),
√
√
φ)] and 2 arctanh[ −c/(2(λi2 ◦ φ))] (respectively arctanh[λi1 ◦
arctanh[ −c/(λi1 ◦ √
√
φ)/ −c]) are distinct at each point of M, then those
φ/ −c] and 2 arctanh[2(λi2 ◦ complex principal curvature functions have constant multiplicities as in Table 2.
In the case where the ambient space is a hyperbolic space, we obtain the following
result in terms of Theorem 5.9.
C OROLLARY 5.11. Let M be an immersed submanifold in the m-dimensional
be as
φ, ṽ, v, A and A
hyperbolic space H m (c) of constant curvature c and let c is
in Theorem 5.9. If the spectrum of Av is {λ1 , . . . , λg }, then the spectrum of A
ṽ
Submanifold geometries in a symmetric space
193
TABLE 2.
Principal curvature function
Constant multiplicity
0
√
−c
∞
√
√
arctanh[ −c/λi ◦ φ] + j π −1
√
−c
√
√
−c] + (j + 1 )π −1
arctanh[λi ◦ φ/
2
√
−c
√
√
2 arctanh[ −c/2(λi ◦ φ)] + 2j π −1
√
−c
√
√
2 arctanh[2(λi ◦ φ)/ −c] + (2j + 1)π −1
dim(D ∩ Ker(Av − λi id))
dim(D ∩ Ker(Av − λi id))
dim(D ⊥ ∩ Ker(Av − λi id))
dim(D ⊥ ∩ Ker(Av − λi id))
given by
√
−c
+
{0} ∪
i∈I , j ∈Z
√
√
arctanh[ −c/λi ] + j π −1 √
−c
i ∈ I −, j ∈ Z
∪
√
√
arctanh[λi / −c] + (j + 12 )π −1 √ √
−c −1 ∪
j ∈ Z \ {0} ,
jπ
√
√ √
where I ± := {i | ±(|λi | − −c) > 0} and −c −1/j π (j ∈ Z \ {0}) are removed
in the case where M is a hypersurface. Furthermore, the multiplicities of the above
c are given as in Table 3.
eigenvalues of A
ṽ
Also, we obtain the following result for a Kaehlerian submanifold in the complex
hyperbolic space in terms of Theorem 5.9.
C OROLLARY 5.12. Let M be an immersed Kaehlerian submanifold in the
m-dimensional complex hyperbolic space CH m (c) of constant holomorphic sectional
be as in Theorem 5.9. If the spectrum of Av is
curvature c and let φ, ṽ, v, A and A
194
N. Koike
TABLE 3.
Eigenvalue
Multiplicity
0
√
−c
∞
√
√
arctanh[ −c/λi ] + j π −1
√
−c
√
√
arctanh[λi / −c] + (j + 12 )π −1
√ √
−c −1
jπ
dim Ker(Av − λi id)
dim Ker(Av − λi id)
codim M − 1
TABLE 4.
Eigenvalue
Multiplicity
0
√
−c
∞
√
√
2 arctanh[ −c/2λi ] + 2j π −1
√
−c
√
√
2 arctanh[2λi / −c] + (2j + 1)π −1
√ √
−c −1
2j π
√ √
−c −1
(2j + 1)π
dim Ker(Av − λi id)
dim Ker(Av − λi id)
codim M − 1
1
c is given by
{λ1 , . . . , λg }, then the spectrum of A
ṽ
√
−c
i ∈ I +, j ∈ Z
{0} ∪
√
√
2 arctanh[ −c/2λi ] + 2j π −1 √
−c
−
i
∈
I
∪
,
j
∈
Z
√
√
2 arctanh[2λi / −c] + (2j + 1)π −1 √ √
−c −1 ∪
j ∈ Z \ {0} ,
jπ
Submanifold geometries in a symmetric space
195
√
where I ± := {i | ±(|λi | − −c/2) > 0}. Furthermore, the multiplicities of the above
c are given as in Table 4.
eigenvalues of A
ṽ
6. Proofs of Theorems A, B, C and D
In this section, we prove Theorems A, B, C and D stated in the introduction. First we
prove Theorem A in terms of Proposition 3.1 and Theorem 5.9.
Let v be a normal vector field of M, v ∗
Proof of Theorem A. Set M ∗ := π(M).
Since M has
(respectively v L ) be the horizontal lift of v to M ∗ (respectively M).
∗
globally flat and abelian normal bundle, we can show that M also has globally flat
and abelian normal bundle and that ∇ ⊥ v = 0 is equivalent to ∇ ∗⊥ v ∗ = 0, where ∇ ⊥
(respectively ∇ ∗⊥ ) is the normal connection of M (respectively M ∗ ) (see the proof of
Lemma 6.3 in [TT]). Furthermore, since M ∗ has abelian normal bundle, it follows
also
⊥ v L = 0 and hence M
from Lemma 5.4 that ∇ ∗⊥ v ∗ = 0 is equivalent to ∇
⊥
has globally flat normal bundle, where ∇ is the normal connection of M. Thus an
is given as the horizontal lift of a parallel
arbitrary parallel normal vector field of M
normal vector field of M. Fix a parallel normal vector field v of M. Also, fix u ∈ M.
Set g0 := φ(u) and x := φ(u). Let {λ1 , . . . , λg } be the set of all principal curvatures
of direction vx . According to Proposition 5.2, there exists g ∈ H 1 ([0, 1], G) satisfying
= g −1 M ∗ . We have g −1 x = eK and
g ∗ u = 0̂, g(0) = g0−1 and φ(g ∗ M)
0
0
−1
= g M. Denote by A, A , A
and A
the shape tensors of M, g −1 M, M
φ(g ∗ M)
0
0
and g ∗ M, respectively. Clearly we have Spec A −1 = Spec Avx = {λ1 , . . . , λg }
g0∗ vx
c
c
and Spec A
(g∗·)∗ vuL = Spec AvuL . Hence, since M is curvature adapted, it follows from
c L is given as
Theorem 5.9 that Spec A
vu
{0} ∪ λi i ∈
α∈
g −1 vx
0∗


∪


∪
α∈+ \
g −1 vx
0∗
α∈+ \
g −1 vx
0∗
Iα
∪{0}
−1
vx )
α(g0∗
√
−1
arctanh[α(g0∗ vx )/λi ] + j π −1
−1
vx )
α(g0∗
−1
arctanh[λi /α(g0∗ vx )] + (j

i ∈ I +, j ∈ Z 

α
√
+ 12 )π −1

i ∈ I −, j ∈ Z 
,
α
(6.1)
196
N. Koike
where + , g −1 vx , ⊥ , Iα and Iα± are as in Theorem 5.9. On the other hand,
0∗
according to Proposition 3.1, it follows from the curvature adaptedness of M that
the set of all complex focal radii of M along γvx consists of the inverse numbers of
elements other than 0 of the set (6.1). These facts deduce statement (i). Furthermore,
−1
vx ) (α ∈ + \ g −1 vx ) are not principal curvatures of direction vx , then
if ±α(g0∗
0∗
it follows from Lemma 5.3 and Propositions 5.6 and 5.7 that there exists a pseudoc L . The converse is also shown.
orthonormal base consisting of the eigenvectors of A
vu
Hence statement (ii) is deduced.
2
Next we prove Theorem C in terms of Lemma 5.3, Propositions 5.6, 5.7 and
Theorem 5.9.
the shape tensor of M (respectively
Proof of Theorem C. Denote by A (respectively A)
Fix a unit normal vector v of M at x and u ∈ M
with M).
φ(u) = x. Set g0 := φ(u).
Let Spec Av = {λ1 , . . . , λg }. Then, in terms of Proposition 5.2 and Theorem 5.9, we
c L is given by the sum of the set (6.1) (where vx is replaced by v) and
see that Spec A
vu
−1 √
the set α∈⊥ \ −1 {α(g0∗
v) −1/(j π) | j ∈ Z \ {0}}. Set mi := dim Ker(Av −
g
0∗
v
λi id), miα := dim(g0∗ pα ∩ Ker(Av − λi id)) and m⊥α := dim(g0∗ pα ∩ Tx⊥ M), where
−1
v) (α ∈ + \ g −1 v ) are not
i = 1, . . . , g and α ∈ + ∪ {0}. Since the α(g0∗
0∗
principal curvatures of direction v by the assumption, it follows from Lemma 5.3
c
and Propositions 5.6 and 5.7 that there exists a pseudo-orthonormal base of Tu M
c
L . For simplicity, set
consisting of the eigenvectors of A
v
u
+
:=
Bαij
−1
α(g0∗
v)
√
−1
arctanh[α(g0∗
v)/λi ] + j π −1
and
−
:=
Bαij
−1
v)
α(g0∗
(i ∈ Iα+ , j ∈ Z)
√
−1
arctanh[λi /α(g0∗
v)] + (j + 12 )π −1
(i ∈ Iα− , j ∈ Z).
Clearly we have
−1
v) coth
λi = α(g0∗
−1
v)
α(g0∗
±
and Bαij
=
−1
±
v)Bαi0
α(g0∗
,
±
−1
± √
Bαi0
α(g0∗
v) + j πBαi0
−1
√
where we have used tanh(θ − π −1/2) = coth θ . These relations together with
coth θ =
1
1
+
√
θ j ∈Z\{0} θ + j π −1
Submanifold geometries in a symmetric space
±
(see the proof of Proposition 5.7) deduce λi = Bαi0
+
g
Tr Av =
mi λi =
g
α∈
g −1 v
0∗
α∈+ \
i∈Iα−
α∈+ \
−1 v
g0∗
miα λi +
∪{0} i∈Iα
+
±
j ∈Z\{0} Bαij .
g −1 v
0∗
i∈Iα+
+
+
miα Bαi0 +
Bαij
lim B ±
j →±∞ αij
j ∈Z\{0}
−1 √
α(g0∗
v) −1
= 0,
= lim
j →±∞
jπ
we have
u
α∈
g −1 v
0∗
+
+
j ∈Z\{0}
m⊥α
α∈+ \
g −1 v
0∗
i∈Iα+
+
+
miα Bαi0
+
Bαij
−
−
miα Bαi0 +
Bαij
g −1 v
0∗
miα λi +
∪{0} i∈Iα
α∈+ \
j ∈Z\{0}
−
−
.
miα Bαi0
+
Bαij
On the other hand, since
c L =
Tr A
v
Hence we obtain
miα λi
i=1 α∈+ ∪{0}
i=1
=
197
i∈Iα−
j ∈Z\{0}
j ∈Z\{0}
√
−1
α(g0∗
v) −1
.
jπ
c L = Tr Av because
Therefore, we obtain Tr A
v
u
−1 √
α(g0∗
v) −1
= 0.
m⊥α
j
π
j ∈Z\{0}
Thus the statement of this theorem follows.
2
Also, we can prove Theorem D in terms of Theorem 5.9.
Proof of Theorem D. Let v be a unit normal vector of M at x. Denote by R the
curvature tensor of N. Since M is curvature adapted, the tangent space Tx M is
R(·, v)v-invariant and Av and R(·, v)v|Tx M are commutative. Since N is of rank
one, the spectrum (considering multiplicities) of R(·, v)v|Tx M is independent of the
choice of v. On the other hand, since M is isospectral, the spectrum (considering
198
N. Koike
multiplicities) of Av is also independent of the choice of v. Furthermore, since Av
and R(·, v)v|Tx M are commutative for each unit normal vector v, the dimensions of
the common-eigenspaces of Av and R(·, v)v|Tx M are independent of the choice of v.
is isospectral.
Therefore, from Theorem 5.9, we can deduce that M
2
Next we prepare two lemmas to prove Theorem B. The following lemma is shown
by imitating the proof of Lemma 6.4 in [TT].
L EMMA 6.1. Let M be an immersed submanifold with globally flat and abelian
normal bundle in a symmetric space N = G/K of non-compact type and let π be
the natural projection of G onto G/K. Let M ∗ be one component of π −1 (M). Let v
be a normal vector field of M and let v ∗ be the horizontal lift of v to M ∗ . Then v
is a focal normal vector field of M with multiplicity k if and only if v ∗ is a focal
normal vector field of M ∗ with multiplicity k. Furthermore, for each g ∈ M ∗ , π∗g
maps the kernel of (ηv∗∗ )∗g bijectively to the kernel of (ηv )∗π(g). Here ηv := exp⊥ ◦ v
(: M → N) and ηv∗∗ := exp∗⊥ ◦v ∗ (: M ∗ → G), where exp⊥ (respectively exp∗⊥ ) is
the normal exponential map of M (respectively M ∗ ).
Also, we obtain the following fact.
L EMMA 6.2. Let M ∗ be an immersed submanifold with globally flat and abelian
normal bundle in a semi-simple non-compact Lie group G and φ be the parallel
be one component of φ −1 (M ∗ ). Let v be a normal vector
transport map for G. Let M
Then v is a focal normal vector
field of M ∗ and ṽ be the horizontal lift of v to M.
∗
with
field of M with multiplicity k if and only if ṽ is a focal normal vector field of M
ηṽ )∗u bijectively
multiplicity k. Furthermore, for each u ∈ M, φ∗u maps the kernel of (
to the kernel of (ηv∗ )∗φ(u) . Here ηv∗ := exp∗⊥ ◦ v (: M ∗ → G) and ηṽ := e
xp⊥ ◦ ṽ
⊥
0
∗⊥
→ H ([0, 1], g)), where exp (respectively e
xp ) is the normal exponential
(: M
∗
map of M (respectively M).
Proof. Since M ∗ has abelian normal bundle, Lemma 5.4 deduces that ∇ ∗⊥ v = 0 is
⊥ ṽ = 0. Also, according to Proposition 4.4, φ is a pseudo-Riemannian
equivalent to ∇
submersion. In terms of these facts, we can show the statement by imitating the proof
of Lemma 5.12 in [TT].
2
Now we prove Theorem B in terms of these lemmas.
is a submanifold in a pseudo-Hilbert space
Proof of Theorem B. Since M
0
is equal to the inverse number of a real principal
H ([0, 1], g), each focal radius of M
curvature of M. By noticing this fact, we can directly deduce the statement in terms
of Lemmas 6.1 and 6.2.
2
Submanifold geometries in a symmetric space
199
7. Concluding remarks
In [T2], Terng defined the notion of the affine coxeter group for an isoparametric
submanifold in a Hilbert space. In this section, we define a similar notion for a
proper complex isoparametric submanifold in a pseudo-Hilbert space. Let M be a
proper complex isoparametric submanifold in a pseudo-Hilbert space (V , ·, ·) and
let {Fi | i ∈ I } (I ⊂ N) be a family of subbundles of T M c such that, for each
x ∈ M, {Fi (x) | i ∈ I } is the set of all common-eigenspaces of Acv (v ∈ Tx⊥ M).
Let µi (respectively wi ) be the complex principal curvature (respectively the complex
curvature normal) corresponding to Fi . Fix x ∈ M. Set li := µ−1
ix (1) (i ∈ I ).
It is shown that the sum i∈I li of the complex hyperplanes li coincides with the
complex focal set of M at x. In the following, assume that wi , wi c = 0 (i ∈ I ).
Then we have Tx⊥ M c = µ−1
ix (0) ⊕ Spanc {wix }. Let ai be the intersection point
of li and Spanc {wix }. Define the affine transformation Ri of Tx⊥ M c by Ri (v) =
prµ−1 (0) (v − ai ) − prwix (v − ai ) + ai (v ∈ Tx⊥ M c ), where prµ−1 (0) (respectively prwix )
ix
ix
is the projection of Tx⊥ M c onto µ−1
ix (0) (respectively Spanc {wix }). We call Ri the
reflection with respect to li . Denote by W the affine transformation group generated
by the Ri (i ∈ I ). It is clear that the group W is essentially independent of the choice
of x ∈ M.
Remark 7.1. In the case where (V , ·, ·) is a Hilbert space (hence M is isoparametric),
each Ri preserves Tx⊥ M and the group generated by the Ri |Tx⊥ M (i ∈ I ) is only
the affine coxeter group (defined by Terng) of the isoparametric submanifold M.
Thus the group W is identified with the affine coxeter group. Hence, it follows from
the properties of the affine coxeter group (see [T2]) that W ( i∈I li ) = i∈I li holds
and that W is discrete.
It is natural to consider the following questions:
Does the relation W ( i∈I li ) = i∈I li hold?
Is the group W discrete?
Question (i) is solved negatively. In fact, we can construct a counter-example
as follows. Denote by M0 the isoparametric hypersurface S n1 (c1 ) × H n2 (c2 ) in the
hyperbolic space H m (c)(= SO0 (m, 1)/SO(m)) (m = n1 + n2 + 1), where S n1 (c1 ) is
the n1 -dimensional sphere of constant curvature c1 and 1/c1 + 1/c2 = 1/c. Let v
be the unit normal vector field of M0 such that the principal curvatures for v are
√
√
c1 − c and c2 − c. The hypersurface M0 is complex equifocal and the above
√
two principal curvatures are distinct from −c. Hence, according to Theorem A,
0 of (π ◦ φ)−1 (M0 ) is proper complex isoparametric, where π is
one component M
(i)
(ii)
200
N. Koike
the natural projection of SO0 (m, 1) onto H m (c) and φ is the parallel transport map
√
√
for SO0 (m, 1). For simplicity, set λ := c1 − c and µ := c2 − c. According to
0 , the spectrum of the shape operator
Corollary 5.11, for the horizontal lift v L of v to M
Av L is given by
√
−c
{0} ∪
√
j ∈Z
√
arctanh[ −c/λ] + j π −1 √
−c
j ∈Z .
∪
√
√
arctanh[µ/ −c] + (j + 12 )π −1 For simplicity, set
λj :=
and
µj :=
√
−c
√
√
arctanh[ −c/λ] + j π −1
√
−c
√
√
arctanh[µ/ −c] + (j + 12 )π −1
(j ∈ Z).
c )∗ defined by ω(v L ) = 1. It is clear that
Let ω be the section of the dual bundle (T ⊥ M
0
0 .
µj ω | j ∈ Z} is the set of all complex principal curvatures of M
{λj ω | j ∈ Z} ∪ {
c
⊥
L
0 . Identify Tu M
with C under vu ↔ 1. Under this identification, we have
Fix u ∈ M
0
λj } (j ∈ Z) and (
µj ωu )−1 (1) = {1/
µj } (j ∈ Z). Hence the above
(
λj ωu )−1 (1) = {1/
affine transformation group W for M0 is regarded as the affine transformation group of
µj (j ∈ Z).
C generated by the rotations of angle π centered on 1/
λj (j ∈ Z) and 1/
It is clear that W is discrete but that W ( j ∈Z {1/
λj , 1/
µj }) j ∈Z {1/
λj , 1/
µj }.
Thus this hypersurface M0 gives a counter-example for question (i).
Therefore, we need to extend the above group W defined for a proper complex
isoparametric submanifold. Let {li }i∈I be as above. Set L1 := {R(li ) | R ∈ W, i ∈ I }.
Let W1 be the affine transformation group generated by the reflections with respect
to elements of L1 . Set L2 := {R(l) | R ∈ W1 , l ∈ L1 }. Let W2 be the affine
transformation group generated by the reflections with respect to elements of L2 .
In the following, we inductively define Li and Wi (i ≥ 3). Clearly we have Li ⊂ Li+1
:= lim→ Wi . We call W
the
and Wi ⊂ Wi+1 (i ≥ 1). Set L := lim→ Li and W
complex affine coxeter group for the proper complex isoparametric submanifold M.
Problem.
(
(i) Does the relation W
l∈
L l) =
l∈
L l hold?
(ii) Is the group W discrete?
Submanifold geometries in a symmetric space
201
for the above
It can be shown that the complex affine coxeter group W
(
0 is discrete and satisfies W
hypersurface M
l∈
L l) =
l∈
L l.
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202
N. Koike
Naoyuki Koike
Department of Mathematics
Faculty of Science
Tokyo University of Science
26 Wakamiya-cho Shinjuku-ku
Tokyo 162-8601
Japan
(E-mail: [email protected])