Graphics Reference
In-Depth Information
Glossary
In the following we provide a brief introduction to the geometrical terms used in the topic.
For details, please refer to some standard textbooks on differential geometry such as Boothby
(1986) or Spivak (1979).
Geodesic Path
: For any two points
p
1
,
connecting
p
1
and
p
2
on
M
that is locally length-minimizing, that is, this path cannot be changed or perturbed
in any way and its length cannot be reduced. If is the shortest path connecting
p
1
and
p
2
then it is called the minimizing geodesic. Note that the definition of a geodesic is dependent
upon the Riemmanian structure of
M
, that is,
p
2
∈
M
, a geodesic is a path
α
•
α
=
ˆ
argmin
α
:[0
,
1]
→
M
,α
(0)
=
p
1
,α
(1)
=
p
2
L
[
α
]
.
The length of this path is called the geodesic length d(
p
1
,
p
2
)
=
L
[ˆ
α
].
Group Action by Isometries
: If a group
G
acts on a Riemannian manifold
M
, then the
group action is said to be by isometries if d(
p
1
,
•
p
2
)
=
d((
g
·
p
1
)
,
(
g
·
p
2
)) for all
g
∈
G
and
p
1
,
p
2
∈
M
.
Group Action on a Manifold
:If
G
is an algebraic group and
M
is a manifold, then the
mapping
G
•
×
M
→
M
, given by (
g
,
p
)
→
g
·
p
is called a group action if: (1) (
g
1
(
g
2
,
p
))
=
(
g
1
∗
g
2
)
·
p
, (2) (
e
,
p
)
=
p
, for all
g
1
,
g
2
∈
G
and
p
∈
M
and where
e
denotes the identity
element of
G
. The set
{
g
·
p
|
g
∈
G
}
is called the orbit of
p
and is denoted by [
p
].
Inherited Distance on
M
G
: As in the previous item, let
G
acts on
M
by isometries and,
additionally, let the orbits of
M
under
G
be closed. Then, the inherited distance between any
two elements of
M
/
•
/
G
is defined to be
d
M
/
G
([
p
1
]
,
[
p
2
])
=
inf
g
G
d
M
(
p
1
,
(
g
·
p
2
))
.
∈
Inherited Metric
: If the action of
G
on a Riemannian manifold
M
is by isometries, and the
quotient space
M
•
G
inherits the Riemannian metric
from
M
and this metric can be used to define geodesics and geodesic distances between
elements of
M
/
G
is a differentiable manifold, then
M
/
/
G
.
Length of Curves
: For any curve
α
:[0
,
1]
→
M
, the length of
α
can be defined as:
•
1
d
(1
/
2)
d
t
,
d
d
t
L
[
α
]
=
d
t
,
0
where the inner product inside the integral is defined using the Riemannian metric of
M
.
Manifold
: A manifold is a topological space that is
locally Euclidean
, that is, for each point
on this set, an open neighborhood can be locally identified with an open set of a Euclidean
space using a homemorphism. If these mappings are smooth (diffeomorphsisms) and com-
patible with each other (their concatenations are also smooth), then the manifold is called
adifferentiable manifold. In this chapter, all the manifolds we deal with are differentiable
manifolds and, hence, we often drop the work differentiable.
•
