DGII Definitions

Question 1

trivial bundle of rank k

Answer 1

.

Question 2

metric space

Answer 2

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Question 3

Inverse Function Theorem for Manifolds

Answer 3

.

Question 4

Push forward

Answer 4

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Question 5

Schwarz Lemma

Answer 5

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Question 6

Maurer-Cartan Lemma

Answer 6

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Question 7

variation

Answer 7

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Question 8

connection

• Levi-civita
• metric
• torsion free
• affine
• curvature

Answer 8

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Question 9

Riemannian curvature tensor

Answer 9

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Question 10

Stoke’s Theorem

Answer 10

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Question 11

wedge product of k and l forms

Answer 11

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Question 12

parallel section

Answer 12

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Question 13

Hopf and Rinow Theorem (Theorem 53)

Answer 13

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Question 14

Second Bianchi Identity

Answer 14

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Question 15

endomorphism

Answer 15

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Question 16

Exponential map

Answer 16

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Question 17

length and energy of curves

Answer 17

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Question 18

pullback bundle

Answer 18

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Question 19

metric

Answer 19

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Question 20

Koszul Formula

Answer 20

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Question 21

exterior derivative

Answer 21

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Question 22

differential of f

Answer 22

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Question 23

Smooth maps

Answer 23

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Question 24

sectional curvature

Answer 24

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Question 25

Lie algebra

Answer 25

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Question 26

Tangent space

Answer 26

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Question 27

First bianchi Identity

Answer 27

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Question 28

vector field

Answer 28

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Question 29

vector bundles

• tensor
• pullback
• flat

Answer 29

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Question 30

vector bundle isomorphism

Answer 30

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Question 31

Atlas

Answer 31

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Question 32

grassmannian

Answer 32

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Question 33

tangent bundle + fiber

Answer 33

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Question 34

differential k-forms

Answer 34

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Question 35

curvature of a connection

Answer 35

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Question 36

Gauss Lemma (Theorem 50)

Answer 36

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Question 37

Compatible Charts

Answer 37

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Question 38

killing fields

Answer 38

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Question 39

diffeomorphism

Answer 39

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Question 40

pullback of

• forms
• connections
• metric

Answer 40

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Question 41

complete Riemannian Manifold

Answer 41

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Question 42

geodesic normal coordinates

Answer 42

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Question 43

section

-parallel

Answer 43

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Question 44

Coordinate frame

Answer 44

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Question 45

isometry

Answer 45

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Question 46

Leibniz’s Rule

Answer 46

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Question 47

bundle-valued differential forms

Answer 47

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Question 48

curvature tensor

Answer 48

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Question 49

torsion tensor/tautological form

Answer 49

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Question 50

submanifold

Answer 50

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Question 51

manifold

• smooth
• topological
• Riemannian
• flat

Answer 51

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Question 52

geodesic

Answer 52

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Question 53

frame field

Answer 53

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Question 54

Ricci Curvature

Answer 54

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Question 55

Scalar curvature

Answer 55

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Question 56

Sectional curvature

Answer 56

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Question 57

M is a complete Riemannian manifold. Then which things are equivalent?

Answer 57

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Question 58

Gauss Bonnet

Answer 58

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Question 59

When is M isometric to a sphere of radius r=1/sqrt(K)?

Answer 59

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Question 60

When is M homeomorphic to Sn?

Answer 60

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