Theory discussed in lectures Riemannian Geometry (MRI, Mastermath)

See also: exercises

Fall semester 2008

(01) Week 37, sep 10

Lecture Notes, Sect. 1: definition and examples of vector bundles;

morphism and isomorphism of vector bundle; trivialization and local frame

skipped: pull-back of vector bundle

sect. 2: connection on a vector bundle in terms of tangent system

skipped: pull-back of connection

horizontal lift of curve and differential equation for it;

affine connection in trivial bundle

(02) Week 38, sep 17

Sect 2: affine connection, horizontal lifting and parallel transport

Sect 2: holonomy group

Sect 3: covariant differentation induced by connection, elementary properties; proof of Lemma 3.2.

(03) Week 39, sep 24

Sect 3: trivialization and local frame; representation of connection by one form, Lemma 3.2.

Sect 3: covariant differential operator always comes from affine connection, Prop. 3.5.

Lemma 3.7: covariant differentiation in terms of parallel transport

Sect 4: definition of curvature R; Lemma 4.3; R as tensor: Cor. 4.4.

Theorem 4.5: curvature as obstruction to flatness. Proof postponed till next week.

(04) Week 40, oct 1

Proof of Thm. 4.5: zero curvature implies flatness connection;

Lecture Notes section 7: Riemannian metrics; Levi-Civita connection.

An extra section 5 on curvature has been added. This will be the subject of a future lecture.

An extra section 6 on a variant of covariant differentiation has been added. This will be the subject of next weeks lecture. Its purpose is to facilitate Exercise 5.10 (now 7.10).

(05) Week 41, oct 8

Section 6: covariant differentiation along a curve.

Section 8: zero curvature for pseudo-Riemannian manifold implies local isometry with flat space

(06) Week 42, oct 15

Section 9: geodesics, geodesic equation, exponential map

Section 10: distance on a Riemannian manifold, up to Lemma 10.2.

(07) Week 43, oct 22

Gauss lemma Lemma 10.2 mentioned, study proof at home

Corollaries 10.3-10.5: application of Gauss lemma: geodesics are curves which are locally length minimalizing.

Sect. 11: Definition of Gauss curvature.

Gauss curvature in terms of curvature tensor.

sectional curvature (see sect. 12).

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(08) Week 44, oct 29

At this point, Stefan Vandoren will start his lecture series. A first set of Lecture Notes is now available.

Lecture Notes II, sect. 1.

(09) Week 45, nov 5

Lecture Notes II, sect. 2

(10) Week 46, nov 12

Lecture Notes II, sect. 3

(11) Week 47, nov 19

Lecture Notes II, sect. 4

(12) Week 48, nov 26

Lecture Notes II, sect. 5

(13) Week 49, dec 3

(14) Week 50, dec 10

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