Crash Course in Riemannian Geometry

Question 1

Define: Riemannian metric

Answer 1

Smooth assignment of inner product to each tangent space

pg 1-2

Question 2

Examples of Riemannian metrics

Answer 2

1. Poincare ball model of H^n
2. Submanifolds and pullback metrics (from immersions)
3. Products
4. Invariant metrics on Lie groups

pg 2-4

Question 3

Define: length of curve, Riemannian distance

Prove this is a metric

Answer 3

Distance = inf {length of curve between pts}

pg 5-6

Question 4

Define: length-minimizing, locally length-minimizing

relation between two?

Answer 4

LM: Distance between endpoints points = length of curve

LLM: In epsilon balls, LM

Question 5

Define: isometry

compare to metric isometry

Answer 5

pullback metric works - infinitesimally preserves distances

metric isometry = distances preserved.

Steenrod. Metric isometry => smooth & Riemannian isometry

Question 6

Define: isometric embedding, totally geodesic embedding, totally geodesic submanifold

Relationships? Examples?

Answer 6

pg 8-10

Question 7

Define: connection, Levi-Civita connection

Answer 7

pg 11

Look up in Lee

Question 8

Define: geodesic

First variation?

Relation to length minimizing and locally length minimizing curves?

Answer 8

pg 12-13

Look up in Lee

Question 9

Discuss finding geodesics - ODE

Answer 9

pg 14

Look up in Lee

Question 10

Define: geodesically complete

examples/non-examples

Answer 10

Every geodesic defined on R

pg 16

Question 11

What is Hopf-RInow Theorem?

Answer 11

geodesically complete <=> metrically complete

pg 17

Question 12

Discuss exponential map. Why called exponential?

Answer 12

pg 17-18

Look up in Lee