ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII “AL.I.CUZA” IAŞI
Tomul XLV, s.I a, Matematică, 1999, f.1.
KILLING POTENTIALS*
BY
MIRCEA CRÂŞMĂREANU
1. Introduction. Let (M, g) be a smooth, finite–dimensional and
connected Riemannian manifold, ∇ its Levi–Civita connection, C ∞ (M ) the
ring of real–valued functions and X (M ) the Lie algebra of smooth vector
fields on M . For f ∈ C ∞ (M ) we denote by ∇f the gradient of f ([3, p. 83])
and by Hf the Hessian of f . Recall that Hf : X (M ) × X (M ) −→ C ∞ (M )
is given by ([3, p. 141–142]):
(1)
Hf (X, Y ) := g(∇X (∇f ), Y )
∀X, Y ∈ X (M )
and X ∈ X (M ) is a Killing vector field if ([3, p.81–82]):
(2)
g(∇Y X, Z) + g(Y, ∇Z X) = 0
∀ Y, Z ∈ X (M ).
We say f ∈ C ∞ (M ) is a Killing potential if ∇f is a Killing vector field.
Then X = ∇f is called a Killing gradient. Killing potentials appear in the
study of some fluids ([5, p. 9]).
2. Properties of Killing potentials. From Hf (X, Y ) = Hf (Y, X)
we have:
(3) 2Hf (X, Y ) = Hf (X, Y ) + Hf (Y, X) = g(∇X (∇f ), Y ) + g(X, ∇Y (∇f ))
which yields the following characterization:
* Dedicated to Professor Dr. Vasile Crucianu on the occasion of his 65th
birthday
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MIRCEA CRÂŞMĂREANU
Theorem. f ∈ C ∞ (M ) is a Killing potential iff Hf = 0
2
(∗).
Applications:
Proposition 1.
(i) Every Killing potential is a harmonic map.
(ii) On a compact, orientable and without boundary Riemannian manifold
the Killing potentials are only constant functions.
Remark. For another formulation of (ii) see Proposition 3.1 of [11,
p. 43].
Proof. (i) From (∗) and relation ∆f := −T rHf ([3, p. 142]).
(ii) From (i) and a theorem of E. HOPF–BOCHNER ([3, p. 85]).
Proposition 2. Let f be a Killing potential.
(i) If N is a regular level of f then N is totally geodesic and |∇f | is constant along each component of N. Suppose that C = {p ∈ M ; df (p) = 0}
is nonempty and let p ∈ C.
(ii) p is degenerate critical point of f .
(iii) There exists a neighborhood V ⊂ M of p such that V ∩ C is a submanifold of M (this implies that every connected component of C is a
submanifold of M ).
Proof. (i) From (∗) and Lemma 1 of [4].
(ii) From (∗) and a very known classification of critical points.
(iii) Ex. 12 b) from [3, p. 143].
Remark. For another formulation of the fact that if f is Killing
potential then |f | is constant see [1, Th.1.].
Proposition 3. Denote by K (M, g) the set of Killing potentials on
(M, g)and by M orse (M ) the set of Morse functions on M . Then:
(i) K (M, g) is submodule in C ∞ (M )
(ii) K (M, g) ⊂ C ∞ (M ) \ M orse (M )
(iii) K (M, g)is a rare set (i. e. int(clK (M, g)) = ∅ where cl=closure and
int=interior) in Whitney WO∞ −topology on C ∞ (M ).
Proof (i) From (∗) .
(ii) From Proposition 2, (ii).
(iii) From (i) and the fact that M orse (M ) is open and dense in
C ∞ (M ) with respect to Whitney WO∞ −topology ([6, p. 147] Th. 1.2.).
Proposition 4. Let f ∈ C ∞ (M ) .
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171
(i) f is Killing potential if and only if ∇f is covariant constant field i.e.
∇X (∇f ) = 0 ∀X ∈ X (M ).
(ii) If f is Killing potential then the orbits of ∇f are geodesics.
Proof. (i) If f is Killing potential then Hf (X, Y ) = g(∇X (∇f ), Y ) = 0,
∀X, Y ∈ X (M ); let Y = ∇X (∇f ). If ∇X (∇f ) = 0 ∀X ∈ X (M ) then Hf = 0.
(ii) In (i) let X = ∇f .
Let f ∈ C ∞ (M ) and hf : X (M ) → X (M ) hf (X) := ∇X (∇f )
∀X ∈ X (M ).
From Hf (X, Y ) = Hf (X, Y ) we have that hf is selfadjoint and then
there exists a orthonormed base (Xi )i=1,n in X (M ) and (di )i=1,n ∈ C ∞ (M )
such that hf (Xi ) = ∇Xi (∇f ) = di Xi i = 1, n. After an ideea of [1] (Th.2,
Th.3) we have:
Proposition 5. f ∈ C ∞ (M ) is Killing potential if and only if di = 0,
i = 1, n.
Proof. Is consequence of Proposition 4, (i).
3. Killing potentials on two dimensional manifolds. By [2] a
∂f ∂f
function f ∈ C ∞ (M ) is called proper if k ∇f k2 = g ij i j 6= 0. For the
∂x ∂x
rest of these paper we work only with proper functions.
In [9] it is proved (using the Brinkmann method [2, p.123]) that there
exists a Killing potential on (M, g) if and only if there is a preferential atlas
on M with respect to which the components of the metric tensor g are
expressed by:
(4)
g11 = K 2
g1α = 0
gαβ = Qαβ (x2 , . . . , xn )
α, β = 2, n
where K is a real number and n is the dimension of M . For improper
functions see [2, p. 131–133].
In the following we obtain all Killing potentials for case n = 2. By (4)
we have
(40 )
g11 = K 2
g12 = g21 = 0 g22 = Q(x2 ).
The metric g is positive definite hence Q(x2 ) > 0 for all x2 . The Christofell
symbols are:
(5)
Γ1ij = 0 Γ211 = Γ212 = 0
Γ222 =
Q0
2Q
i, j = 1, 2
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MIRCEA CRÂŞMĂREANU
4
where Q0 denote the derivative of Q. Locally, the condition (∗) is expresssed
as follows:
∂2f
∂f
− Γkij k = 0
i
j
∂x ∂x
∂x
(6)
i, j = 1, 2.
Using (5) and (6) we have:
∂2f
=0
∂(x1 )2
∂2f
Q0 ∂f
=
·
∂(x2 )2
2Q ∂x2
∂2f
=0
∂x1 ∂x2
∂f
From the first two equations it results
= C1 and from the last
∂x1
p
∂f
= C2 Q where C1 , C2 are real numbers. It follows:
2
∂x
Proposition 6. On a two dimensional Riemannian manifold with
respect to atlas who define (4) the Killing potentials are given by:
1
2
1
Z
f (x , x ) = C1 x + C2
x2
p
Q(t)dt + C3
0
where C1 , C2 , C3 are real numbers. The asociate Killing gradients are:
X=
C1 ∂
∂
C2
+p
·
2
K 2 ∂x1
∂x
Q(x ) 2
Remark. We observe that kXk2 =
with Proposition 2, (i).
C12
+ C22 =constant in agreement
K2
Let Vn be a Riemannian manifold with the components of metric tensor:
gij = aij (x1 , . . . , xm ), giα = 0, gαβ = ψ(x1 , . . . , xm )bαβ (xm+1 , . . . , xn )
i, j = 1, m, α, β = m + 1, n where aij are the components of the metric tensor
for a Vm -Riemannian manifold and bαβ are the components of the metric
tensor for a Vn−m -Riemannian manifold. If Vn−m is of constant curvature
then Vn is called a subprojectif manifold of (n-m-1)-order ([10, p. 41]). For
n = 2, m = 1, aij (x1 ) = K 2 , ψ(x1 ) = 1, bαβ (x2 ) = Q(x2 ) we obtain the
relations (40 ) and the curvature tensor of (V2−1 , bαβ = Q) is zero. Then:
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173
Proposition 7. A two dimensional Riemannian manifold on which
there exists Killing potentials is a subprojectif manifold of 0-order.
Returning to general case (4), remark that (Qαβ ) is a Riemannian
α
metric for a Vn−1 -manifold. We obtain Γ111 =Γ11α =Γ1αβ =Γα
11 =Γ1β =0 and
α
2
n
Γα
βρ = Γβρ (x , . . . , x ) are the Christoffel symbols for (Vn−1 , Qαβ ). From:
(7)
∂2f
=0
∂(x1 )2
∂2f
=0
∂x1 ∂xα
∂2f
∂f
= Γραβ ρ
α
β
∂x ∂x
∂x
∂f
= C1 and for every x1 the function
∂x1
(x2 , ..., xn ) −→ f (x1 , x2 , ..., xn )
is Killing potential for (Vn−1 , Qαβ ).
it follows that
4. Killing potentials with respect to a pair of Riemannian
metrics. Two Riemannian metrics g, g̃ on M is called in g-subgeodesic correspondence ([7]) if there exists ξ = ξ i i=1,n ∈ T01 (M ) and ψ = (ψi )i=1,n ∈
∈ T10 (M ) such that:
(8)
Γ̃ijk = Γijk + δji ψk + δki ψj + gjk ξ i
Proposition 8. Let g, g̃ two Riemannian metrics on M in g-subgeodesic correspondence and f ∈ C ∞ (M ) proper. If f is Killing potential with
respect to both g, g̃ then ψ ≡ 0 and there is a preferential atlas on M with
respect to which we have
(9)
2 α
α
ξ 1 = 0, Γ̃α
11 = K ξ , Γ̃1β = 0, α, β = 2, n
with K 6= 0 a real number.
Proof. By [2 p.123] there is a preferential atlas on M with respect to
which f = x1 , g1α = g 1α = 0, α = 2, n. Then g is given by (4) and then it
result (7). We have:
∂2f
1
1
2 1
2 = 0 ⇒ Γ̃11 = Γ11 = 0 ⇒ (a) 2ψ1 + K ξ = 0
1
∂ (x )
∂2f
= 0 ⇒ Γ̃11α = Γ11α = 0 ⇒ (b) ψα = 0
∂x1 ∂xα
∂2f
= 0 ⇒ Γ̃1αβ = Γ1αβ = 0 ⇒ gαβ ξ 1 = 0 ⇒ (c) ξ 1 = 0
∂xα ∂xβ
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MIRCEA CRÂŞMĂREANU
6
(a)+(c) ⇒ ψ1 = 0 and then ψ = 0.
α
2 α
Γα
11 = 0 ⇒ (d) Γ̃11 = K ξ
α
α
Γ1β = 0 ⇒ (e) Γ̃1β = 0.
Particular cases:
1. If ξ = 0 in (8) the metrics g, g̃ are called in geodesic correspondence
([10, p. 322]) Then:
Proposition 9. Let g, g̃ two Riemannian metrics on M in geodesic
correspondence and f ∈ C ∞(M ) proper.
If f is Killing potential with respect to both g, g̃ then ψ = 0 i.e.Γ̃ = Γ and there is a preferential atlas on
M with respect to which we have (7).
2. Let g, g̃ = e2u g with u ∈ C ∞ (M ) two conformal metrics on M .
Then:
∂u
∂u
∂u
Γ̃ijk = Γijk + δji k + δki j − gjk g im m
∂x
∂x
∂x
i.e. (8) with ψ = du, ξ = −∇u where gradient is with respect to g. Then:
Proposition 10. Let g and g̃ = eu g, u ∈ C ∞ (M ) two conformal
Riemannian metrics on M and f ∈ C ∞ (M ) proper. If f is Killing potential
with respect to both g and g̃ then ψ = du = 0 i.e. u is constant.
5. Lifts on T M . Let G be the Sasaki lift of g to T M . By (∗) a
function f˜ ∈ C ∞ (T M ) is Killing potential on (T M, G) if and only if:
˜
˜
∂ 2 f˜
n+k ∂ f
k ∂f
−
Γ̃
−
Γ̃
=0
ij
ij
∂xi ∂xj
∂xk
∂y k
˜
∂ 2 f˜
∂ f˜
n+k ∂ f
k
−
Γ̃
−
Γ̃
=0
Bij (f˜) :=
in+j
in+j
∂xi ∂y j
∂xk
∂y k
∂ 2 f˜
∂ f˜
∂ f˜
n+k
k
Cij (f˜) :=
−
Γ̃
−
Γ̃
=0
n+in+j
n+in+j
∂y i ∂y j
∂xk
∂y k
Aij (f˜) :=
where (Γ̃) are the Christoffel symbols for G ([8, p. 194]).
Let f ∈ C ∞ (M ) be a Killing potential on the base manifold M and
the lifts on T M :
(i) the vertical lift: f V := f ◦ π where π : T M −→ M is the natural
projection
∂f i
(ii) the complet lift: f C =
y
∂xi
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KILLING POTENTIALS
175
(iii) the horizontal lift: f H = 0.
In the following the Greek indices λ, µ, . . . range over 1, 2, . . . , n and
i
(R = Rjkl
) is the curvature tensor of M . After a straightforward computation we obtain for f V :
1 k
∂f
k
AVij = − (Rjhµ
Γhλi + Rihµ
Γhλj )y λ y µ k
2
∂x
1 k λ ∂f
V
Bij
= − Rijλ
y
2
∂xk
V
Cij
= 0.
V h
V h
Remark that AVij = (Bjh
Γλi + Bih
Γλj )y λ which yields:
Proposition 11. If f is a Killing potential on M then f V is Killing
k ∂f
potential on (T M, G) if and only if Rijl
= 0 i, j, l = 1, n or globally
∂xk
df ◦ R = 0 where df is the differential of f .
For f C we obtain:
1 k
k
λ ∂f
AC
ij = − (Riλj + Rjλi )y
2
∂xk
C
C
Bij
= Cij
=0
and then:
Proposition 12. If f is a Killing potential on M then f C is Killing
∂f
k
k
potential on (T M, G) if and only if (Rilj
+ Rjli
) k = 0 i, j, l = 1, n or
∂x
globally df (R(X, Y )Z + R(X, Z)Y ) = 0 ∀X, Y, Z ∈ X (M ).
6. Classification of Killing potentials. In this section we provide
a definitive solution to the following problem.
Describe the complete connected Riemann manifolds (M, g) which admit nontrivial Killing potentials.
We will prove the following result.
Classification Theorem. A complete, connected Riemann manifold
(M, g) which admits a nontrivial Killing potential is isometric to a product
R×N . The Killing potential in this case is the natural projection R×N →R.
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MIRCEA CRÂŞMĂREANU
8
Proof. If f is a nontrivial Killing potential then ∇f is not identically
0. Condition (i) by Proposition 4 above implies (since M is connected)
that ∇f never vanishes. Since the flow lines of ∇f are geodesics and M
is complete the flow is defined for all moments of time. Then we have the
following isometry (since the flow generated by ∇f is a flow of isometries)
M∼
= R × f −1 (0).
The theorem is proved.
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Received: 7.II.1997
Faculty of Mathematics
”Al.I. Cuza” University
R-6600 Iaşi, ROMANIA
e-mail address: [email protected]