Graphics Reference
In-Depth Information
Figure 9.28.
The exponential map.
T p (S)
tv
v
p
S
exp p (tv)
exp p (v)
and define a map
() Æ
() =
()
exp
:
Up
S
by
exp
v
g 1
.
p
p
See Figure 9.28.
Definition.
The map exp p is called the exponential map of the surface S at p .
The exponential map is defined in a neighborhood of the origin of each tangent
space. If the surface is geodesically complete, then the exponential map is defined on
all of T p ( S ). The exponential maps exp p at points p define a map
open neighborhood of zero cross - section
in total space of tangent bundle t S
Ê
Ë
ˆ
¯
exp:
Æ
S
that is also called the exponential map .
9.10.21. Theorem.
(1) Both exp p and exp are differentiable maps.
(2) The map exp p is a diffeomorphism of a neighborhood of the origin in T p ( S )
onto a neighborhood of p in S .
(3) For each p ΠS and v ΠU ( p ), the unique geodesic g from p to q = exp p ( v ) is
defined by
() =
()
g t
exp
t
v
.
p
Proof.
See [Thor79].
We see from Theorem 9.10.21 that the exponential map essentially formalizes the
concept of an “orthogonal projection” of a neighborhood of the origin in the tangent
space T p ( S ) to S .
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