DGII Definitions Flashcards
1
Q
trivial bundle of rank k
A
.
2
Q
metric space
A
.
3
Q
Inverse Function Theorem for Manifolds
A
.
4
Q
Push forward
A
.
5
Q
Schwarz Lemma
A
.
6
Q
Maurer-Cartan Lemma
A
.
7
Q
variation
A
.
8
Q
connection
- Levi-civita
- metric
- torsion free
- affine
- curvature
A
.
9
Q
Riemannian curvature tensor
A
.
10
Q
Stoke’s Theorem
A
.
11
Q
wedge product of k and l forms
A
.
12
Q
parallel section
A
.
13
Q
Hopf and Rinow Theorem (Theorem 53)
A
.
14
Q
Second Bianchi Identity
A
.
15
Q
endomorphism
A
.
16
Q
Exponential map
A
.
17
Q
length and energy of curves
A
.
18
Q
pullback bundle
A
.
19
Q
metric
A
.
20
Q
Koszul Formula
A
.
21
Q
exterior derivative
A
.
22
Q
differential of f
A
.
23
Q
Smooth maps
A
.
24
Q
sectional curvature
A
.
25
Lie algebra
.
26
Tangent space
.
27
First bianchi Identity
.
28
vector field
.
29
vector bundles
- tensor
- pullback
- flat
.
30
vector bundle isomorphism
.
31
Atlas
.
32
grassmannian
.
33
tangent bundle + fiber
.
34
differential k-forms
.
35
curvature of a connection
.
36
Gauss Lemma (Theorem 50)
.
37
Compatible Charts
.
38
killing fields
.
39
diffeomorphism
.
40
pullback of
- forms
- connections
- metric
.
41
complete Riemannian Manifold
.
42
geodesic normal coordinates
.
43
section
| -parallel
.
44
Coordinate frame
.
45
isometry
.
46
Leibniz's Rule
.
47
bundle-valued differential forms
.
48
curvature tensor
.
49
torsion tensor/tautological form
.
50
submanifold
.
51
manifold
- smooth
- topological
- Riemannian
- flat
.
52
geodesic
.
53
frame field
.
54
Ricci Curvature
.
55
Scalar curvature
.
56
Sectional curvature
.
57
M is a complete Riemannian manifold. Then which things are equivalent?
.
58
Gauss Bonnet
.
59
When is M isometric to a sphere of radius r=1/sqrt(K)?
.
60
When is M homeomorphic to Sn?
.
