Title
Parallel Transport and Geodesics
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Readings
Readings
Readings: Section 4.4 (but not Lemma 1, Proposition 3 and
Proposition 4)
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Vector field on an open set
Definition
Let U be an open subset of the surface S. A tangent vector field on
U is an assignment of a vector w(p) in Tp S to each point p in S.
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Vector field on an open set
Definition
Let U be an open subset of the surface S. A tangent vector field on
U is an assignment of a vector w(p) in Tp S to each point p in S.
Two ways to think of this:
• as a collection of vectors tangent to the surface ranging over U
• as a map w : U → R3 (where w(p) is in Tp S)
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Vector field on an open set
Definition
Let U be an open subset of the surface S. A tangent vector field on
U is an assignment of a vector w(p) in Tp S to each point p in S.
Two ways to think of this:
• as a collection of vectors tangent to the surface ranging over U
• as a map w : U → R3 (where w(p) is in Tp S)
You’ll learn more about vector fields in your vector calculus course.
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Vector field along a curve
Definition
Given a parametrized curve α : I → S in a regular surface S a vector
field along α is an assignment of a vector w(t) in Tw(t) S to each t in I.
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Vector field along a curve
Definition
Given a parametrized curve α : I → S in a regular surface S a vector
field along α is an assignment of a vector w(t) in Tw(t) S to each t in I.
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Differentiability of vector fields on a curve
Definition
We say that α : I → S is differentiable at t0 if for some parametrization
x(u, v ) of the surface S, the coefficients a and b in the expression
w = axu + bxv
are differentiable at t0 . If w is differentiable at all t in I then we say w is
differentiable on I.
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Differentiability of vector fields on a curve
Definition
We say that α : I → S is differentiable at t0 if for some parametrization
x(u, v ) of the surface S, the coefficients a and b in the expression
w = axu + bxv
are differentiable at t0 . If w is differentiable at all t in I then we say w is
differentiable on I.
• in full, w(t) = a(t)xu (u(t), v (t)) + b(t)xv (u(t), v (t)) (where
α(t) = x(u(t), v (t))).
• If a and b are differentiable in one parametrization, then the
coefficients a0 and b0 in another parametrization are also differentiable.
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Derivative at end point
Technical point: In this section we allow parametrized curves to be
defined on closed intervals: α : [a, b] → S.
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Derivative at end point
Technical point: In this section we allow parametrized curves to be
defined on closed intervals: α : [a, b] → S.
By this we mean the restriction to [a, b] of a parametrized curve
α : (a − , b + ) → S.
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Derivative at end point
Technical point: In this section we allow parametrized curves to be
defined on closed intervals: α : [a, b] → S.
By this we mean the restriction to [a, b] of a parametrized curve
α : (a − , b + ) → S.
You can talk about derivatives at end-points by either taking left or right
hand derivatives, or differentiating the extended curve on (a − , b + ).
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Derivative at end point
Technical point: In this section we allow parametrized curves to be
defined on closed intervals: α : [a, b] → S.
By this we mean the restriction to [a, b] of a parametrized curve
α : (a − , b + ) → S.
You can talk about derivatives at end-points by either taking left or right
hand derivatives, or differentiating the extended curve on (a − , b + ).
There are some fine points but I won’t try to trip you up on them.
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Orthogonal projection onto the tangent plane
Definition
Let S be a regular surface and p be a point in S. Let Pp : R3 → Tp S be
the orthogonal projection onto the tangent plane Tp S.
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Covariant derivative of a vector field on a curve
Definition
Let w(t) be a tangent vector field defined along a parametrized curve
α : (a, b). Let c be in the interval (a, b). The covariant derivative of w
at p = α(c) is
Dw
dw
(c) = Pp
(c).
dt
dt
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Covariant derivative of a vector field on a curve
Definition
Let w(t) be a tangent vector field defined along a parametrized curve
α : (a, b). Let c be in the interval (a, b). The covariant derivative of w
at p = α(c) is
Dw
dw
(c) = Pp
(c).
dt
dt
Idea of covariant derivative: it measures the derivative of the vector
field in the surface.
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Formula for covariant derivative
The covariant derivative can be computed in terms of the Christoffel
symbols.
Assume that
w(t) = a(t) xu (u(t), v (t)) + b(t) xv (u(t), v (t)).
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Formula for covariant derivative
The covariant derivative can be computed in terms of the Christoffel
symbols.
Assume that
w(t) = a(t) xu (u(t), v (t)) + b(t) xv (u(t), v (t)).
Then
Dw 0
= a + Γ111 au 0 + Γ112 av 0 + Γ112 bu 0 + Γ122 bv 0 xu
dt
+ b0 + Γ211 au 0 + Γ212 av 0 + Γ212 bu 0 + Γ222 bv 0 xv .
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A useful theorem
Theorem
Let S and S be oriented surfaces. Let φ : U → V be an isometry
between U ⊆ S and V ⊆ S.
Let α : I → U be a curve in U and β(t) = φ ◦ α(t) the image in V . Let
w(t) be a vector field along α and v(t) = dφα(t) (w(t)) be its image
under φ. Then
Dv
Dw
=
dt
dt
for all t.
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A useful theorem
Theorem
Let S and S be oriented surfaces. Let φ : U → V be an isometry
between U ⊆ S and V ⊆ S.
Let α : I → U be a curve in U and β(t) = φ ◦ α(t) the image in V . Let
w(t) be a vector field along α and v(t) = dφα(t) (w(t)) be its image
under φ. Then
Dv
Dw
=
dt
dt
for all t.
Short version: Isometries don’t change the covariant derivative.
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Parallel Vector Fields
Definition
A vector field w along a parametrized curve α : I → S is called parallel
if
Dw
=0
dt
for all t in I.
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Parallel Vector Fields
Definition
A vector field w along a parametrized curve α : I → S is called parallel
if
Dw
=0
dt
for all t in I.
Proposition
If w(t) and v(t) are parallel along a curve α : I → S then
hv(t), w(t)i = constant.
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Parallel Vector Fields
Definition
A vector field w along a parametrized curve α : I → S is called parallel
if
Dw
=0
dt
for all t in I.
Proposition
If w(t) and v(t) are parallel along a curve α : I → S then
hv(t), w(t)i = constant.
• In particular, the angle between v and w is constant
• if w is parallel, then its length is constant.
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Parallel transport
Proposition
Given a parametrized curve α : I → S and a vector w0 in Tα (t0 )S,
there is a unique parallel vector field w on α such that w(t0 ) = w0 .
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Parallel transport
Proposition
Given a parametrized curve α : I → S and a vector w0 in Tα (t0 )S,
there is a unique parallel vector field w on α such that w(t0 ) = w0 .
• We call this vector field the parallel transport of w0 along α.
• the parallel transport does not depend on the parametrization.
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Geodesics and geodesic curvature
Geodesics
Definition
A nonconstant parametrized curve α : I → S is a called a geodesic at
t0 in I if
Dα0
(t0 ) = 0.
dt
It is called a parametrized geodesic if it is a geodesic at t for all t in I.
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Geodesics and geodesic curvature
Geodesics
Definition
A nonconstant parametrized curve α : I → S is a called a geodesic at
t0 in I if
Dα0
(t0 ) = 0.
dt
It is called a parametrized geodesic if it is a geodesic at t for all t in I.
• In other words, its tangent vector is parallel.
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Geodesics and geodesic curvature
Geodesics
Definition
A nonconstant parametrized curve α : I → S is a called a geodesic at
t0 in I if
Dα0
(t0 ) = 0.
dt
It is called a parametrized geodesic if it is a geodesic at t for all t in I.
• In other words, its tangent vector is parallel.
A regular curve is called a geodesic if it is a parametrized geodesic
with respect to its arc length parametrization.
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Geodesics and geodesic curvature
Equations for geodesic
Let α : I → S be a curve in S. If α(t) = x(u(t), v (t)) then the tangent
vector can be written:
α0 (t) = u 0 (t)xu + v 0 (t)xv .
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Geodesics and geodesic curvature
Equations for geodesic
Let α : I → S be a curve in S. If α(t) = x(u(t), v (t)) then the tangent
vector can be written:
α0 (t) = u 0 (t)xu + v 0 (t)xv .
Then α is a geodesic if and only if
u 00 + Γ111 (u 0 )2 + 2Γ112 u 0 v 0 + Γ122 (v 0 )2 = 0
v 00 + Γ211 (u 0 )2 + 2Γ212 u 0 v 0 + Γ222 (v 0 )2 = 0.
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Geodesics and geodesic curvature
Geodesic curvature
We say a curve is oriented if it has an assigned direction to be traced.
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Geodesics and geodesic curvature
Geodesic curvature
We say a curve is oriented if it has an assigned direction to be traced.
If α is parametrized by arc length then α00 is perpendicular to α0 . So
Dα0
0
dt is perpendicular to α and N.
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Geodesics and geodesic curvature
Geodesic curvature
We say a curve is oriented if it has an assigned direction to be traced.
If α is parametrized by arc length then α00 is perpendicular to α0 . So
Dα0
0
dt is perpendicular to α and N.
Definition
Let C be an oriented regular curve on an oriented surface S and let p
be a point on C. Let α(s) be a parametrization by arc length of C.
Define the geodesic curvature kg (p) of C at p by
Dα0
(s0 ) = kg (p)N(s0 ) ∧ α0 (s0 ).
dt
where α(s0 ) = p.
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Geodesics and geodesic curvature
Interpretation of geodesic curvature
Let α be a regular curve, parametrized by arc length, in an oriented
surface S.
Let w(s) be any parallel vector field along α.
Let φ be the angle between α0 (s) and w (chosen so that it varies
continuously along α).
The geodesic curvature of α at p = α(s0 ) is
kg (s0 ) =
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dφ
.
ds
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Geodesics and geodesic curvature
A few words on geodesics
1
2
They are curves which are “as straight as possible” without
leaving the surface.
The “bending” of a geodesic (the second derivative α00 , assuming
arc length parametrization) is perpendicular to the surface.
3
The geodesic curvature of a geodesic is zero.
4
It can be shown that the shortest curve between two given points
is a geodesic.
(Some of these statements are different ways of saying the same
thing).
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