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
Œ
S
and
v
Œ
U
(
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
.