Riemannian Geometry IV

MATH4171

Lecturer : Norbert Peyerimhoff

Terms : Michaelmas 2010, Epiphany 2011

Lectures :
  • Mondays, 11:00 in CG 83
  • Mondays, 16.15 in CG 83

Problem Classes :

  • Friday, 5 November, 3.15 in W205
  • Thursday, 10 February, 2.15 in E005

Literature

The following is a list of books on which the lecture is based. Although we will not follow a book strictly, the material can be found in them and they may sometimes offer a different approach to the material.
  • M. Do Carmo, Riemannian Geometry. Birkhaeuser Verlag.
  • S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry. Springer Verlag.
  • J. Lee, Riemannian manifolds: an introduction to curvature. Springer Verlag.


Assignments

Homework Date Hand in Solutions
Exercise Sheet 1 pdf 11.10.2010 --- Solution Sheet 1 pdf
Exercise Sheet 2 pdf 18.10.2010 25.10.2010 Solution Sheet 2 pdf
Exercise Sheet 3 pdf 25.10.2010 --- Solution Sheet 3 pdf
Exercise Sheet 4 pdf 1.11.2010 --- Solution Sheet 4 pdf
Exercise Sheet 5 pdf 8.11.2010 --- Solution Sheet 5 pdf
Exercise Sheet 6 pdf 15.11.2010 --- Solution Sheet 6 pdf
Exercise Sheet 7 pdf 22.11.2010 --- Solution Sheet 7 pdf
Exercise Sheet 8 pdf 29.11.2010 6.12.2010 Solution Sheet 8 pdf
Exercise Sheet 9 pdf 6.12.2010 --- Solution Sheet 9 pdf
Exercise Sheet 10 pdf 13.12.2010 --- Solution Sheet 10 pdf
Exercise Sheet 11 pdf 17.1.2011 --- Solution Sheet 11 pdf
Exercise Sheet 12 pdf 24.1.2011 31.1.2011 Solution Sheet 12 pdf
Exercise Sheet 13 pdf 31.1.2011 --- Solution Sheet 13 pdf
Exercise Sheet 14 pdf 7.2.2011 --- Solution Sheet 14 pdf
Exercise Sheet 15 pdf 14.2.2011 --- Solution Sheet 15 pdf
Exercise Sheet 16 pdf 21.2.2011 --- Solution Sheet 16 pdf
Exercise Sheet 17 pdf 28.2.2011 --- Solution Sheet 17 pdf
Exercise Sheet 18 pdf 28.2.2011 --- Solution Sheet 18 pdf


Content of Lectures

Date Content
11 October 2010, 11am review of results of Differential Geometry (curves and surfaces in R^3, first fundamental form, geodesics, curvatures and ambient space) and possible generalisations; concept of an abstract differentiable manifold (without ambient space)
11 October 2010, 4.15pm examples of manifolds and non-manifolds (curves, surfaces, projective spaces, violation of Hausdorff property); manifolds and regular values (preimage of regular value is manifold, dimension of this manifold); matrix groups as manifolds (e.g. O(n), SL(n))
18 October 2010, 11am SO(n) as manifold (by using regular value approach), vector space of differentiable functions on manifold, Lie groups, curves and directional derivatives, directional derivatives induces an equivalence relation on curves
18 October 2010, 4.15pm Linear derivations, tangent vectors and tangent space, tangent space as vector space and basis partial/partial{x_i} with respect to coordinate chart, tangent space in the special case of submanifold in R^N, formula for the derivative of the determinant of a curve in the space of matrices (see slides)
25 October 2010, 11am Tangent space of SL(n,R) as application of derivative formula for determinant, Lie groups GL(n,R),SL(n,R),O(n),S)(n) and their tangent spaces, homogeneous spaces as manifolds, dimension of Grassmannians and their tangent spaces, definition of the differential
25 October 2010, 4.15pm Example of a differential, Whitney embedding theorem, Tangent bundle as manifold and footpoint projection, vector fields, space of vector fields as vector space
1 November 2010, 11am Examples of vector fields, vector fields as 1st order differential operators on space of differentiable functions, Lie bracket of vector fields, Lie bracket in local coordinates, properties of the Lie bracket
1 November 2010, 4.15pm Riemannian metric, examples of Riemannian manifolds (Euclidean space, surfaces), connection betwwen Riemannian metric and first fundamental form in differential geometry, lenght of tangent vector, hyperboloid model of the hyperbolic space
8 November 2010, 11am Poincare model and upper half space model of the hyperbolic space, isometries, example of an isometry, group of isometries as subgroup of the diffeomorphism group, group action on a set, Riemannian homogeneous manifold
8 November 2010, 4.15pm Hyperbolic plane as Riemannian homogeneous manifold, integration on Riemannian manifold, independence of the integration formula under change of coordinates, volume of a subset of a Riemannian manifold, example of a volume calculation, zero sets, partition of unity
15 November 2010, 11am globalisation of the integration for manifolds with more than one coordinate chart, Nash Embedding Theorem, length of curve, invariance under reparametrisation, arc-length parametrisation, Example of the length of a curve in hyperbolic space
15 November 2010, 4.15pm distance of two points in connected Riemannian manifold, compactness, completeness, covariant derivative as directional derivative of a vector field, properties of the covariant derivative
22 November 2010, 11am continuation of properties of the covariant derivative, Theorem of Levi-Civita about existence and uniqueness of a Riemannian and torsion free connection
22 November 2010, 4.15pm Levi-Civita connection for surfaces in R^3, Definition and explicit calculation of Christoffel symbols, Gamma_{ij}^k = Gamma_{ji}^k
29 November 2010, 11am Christoffel symbols determine Levi-Civita connection, example of covariant derivative of a vector field in the hyperbolic plane, covariant derivative along a curve, example of covariant derivative on surfaces in R^3 and re-interpretation of geodesics
29 November 2010, 4.15pm Parallel vector fields, example of parallel vector fields in R^k, Theorem on uniqueness of the parallel vector field for given inital vector, Example of parallel transport along horizontal curve in the hyperbolic plane
6 December 2010, 11am Continuation of an example of parallel transport along horizontal curve in the hyperbolic plane, parallel transport, Levi-Civita connection induces a connection between disjoint tangent spaces via parallel transport along a curve connecting their footpoints, recap of facts about matrix Lie groups
6 December 2010, 4.15pm Left-invariant vector fields, Lie algebra as finite vector space, Lie bracket of left-invariant vector fields is again left-invariant, Lie group exponential map for matrix Lie groups
13 December 2010, 11am Properties of Exp(tv), 1-parameter groups, exponential map of diagonisable matrices, abstract definition of the Lie group exponential map, adjoint representations Ad(g) and ad(v), relation between ad and the Lie bracket
13 December 2010, 4.15pm Example of Lie bracket for matrix Lie groups, left-invariant Riemannian metrics, existence of bi-invariant Riemannian metrics in case of compact Lie groups, identity <[X,Y],Z=- in case of a bi-invariant metric, expressing Levi-Civita connection as Lie bracket in case of a bi-invariant metric
17 January 2011, 11am Repetition of left- and right-invariant metrics on Lie groups, left-, right- and bi-invariant Haar measures and transformation rule, invariant metrics an homogeneous spaces G/H, relation between those invariant metrics and Ad(H)-invariant inner products on the tangent space at the identity of G
17 January 2011, 4.15pm Geometric and physical properties of straight lines, definition of geodesic, geodesics are parametrised proportional to arc length, uniqueness of geodesic for given start vector, Examples: geodesics on spheres and in the hyperbolic plane
24 January 2011, 11am Continuation of Example: geodesics in the hyperbolic plane, definition of a variation and of a variational vector field, definition of length and energy, characterisation of geodesics as critical points of the length functional, consequence that distance minimizing curves parametrised proportional to arc length are geodesics
24 January 2011, 4.15pm Symmetry lemma, first variation formula of length and its proof using crucially the symmetry lemma, application to characterisation of geodesics as critical points of the length functional
31 January 2011, 11am Rescaling lemma of geodesics, geodesic flow as dynamical system, acting on the tangent bundle, identification of the tangent space at v of TM with the cartesian product of two copies of the tangent spaces of M at the footpoint of v, horizontal and vertical subspaces of this tangent space, definition of the Riemannian exponential map
31 January 2011, 4.15pm Example: exponential map on 2-sphere and on 2-sphere with antipodal point removed, result that restriction of exponential map to small enough epsilon-ball in tangent space is a diffeomorphism on its image, Gauss-Lemma and implications
7 February 2011, 11am Definition of minimal geodesics, Definition of geodesically complete Riemannian manifolds, Examples to illustrate of both notions, Theorem of Hopf-Rinow, Strategy for how to find minimal geodesics for given two points
7 February 2011, 4.15pm Definition of Riemannian Curvature Tensor, tensorial property of Riemannian Curvature Tensor, symmetry properties and their proofs, Riemannian Curvature Tensor in example of hyperbolic upper half plane
14 February 2011, 11am Continuation of Riemannian Curvature Tensor in example of hyperbolic upper half plane, derivation of formula for sectional curvature, denominator describes area of parallelogram spanned by the two vectors, connection of sectional curvature and Gaussian curvature via taking local image of two-dimensional subspace of tangent plane under exponential map
14 February 2011, 4.15pm Example: calculation of some sectional curvatures in three dimensional hyperbolic upper half plane and corresponding two dimensional submanifolds, isometry explanation for constant sectional curvature -1, concrete form of Riemannian Curvature Tensor in case of constant sectional curvature, trace of an endomorphism of a vector space with inner product, Ricci Curvature Tensor and Ricci Curvature, symmetry of the Ricci Curvature Tensor, definition of an Einstein manifold, definition of scalar curvature
21 February 2011, 11am Second variation formula of length, formulation of the Theorem of Bonnet-Myers, Example of the round sphere of radius r, which shows that the result of Bonnet-Myers is optimal, begin of proof of Bonnet-Myers
21 February 2011, 4.15pm Continuation of proof of Bonnet-Myers, discussion how Bonnet-Myers implies that an n-dimensional torus does not admit a metric of strictly positive Ricci curvature, explanation how Gauss-Bonnet from Differential Geometry implies the same result for tori of dimension two
28 February 2011, 11am Recalling facts about variations, in particular the Symmetry Lemma, Lemma that commutator of the two covariant derivatives of a vector field defined in connection with a variation leads to curvature expression, Second Variation Formula of Energy with proof, definition of the index form
28 February 2011, 4.15pm Cancelled
7 March 2011, 11am Definition of Jacobi fields, Jacobi fields as variational vector fields of geodesic variations, Special examples of Jacobi fields, Example of a Jacobi field on a two-sphere, proof of uniqueness of Jacobi fields for giving initial conditions
7 March 2011, 4.15pm Dimension of space of Jacobi fields along a geodesic, definition of conjugate point, multiplicity of a conjugate point, Example of a conjugate point on the round two-sphere, orthogonal Jacobi fields and dimensional considerations
14 March 2011, 11am Connections between conjugate points and critical points of the exponential map, definition of a simply connected space, examples of simply connected spaces, the Theorem of Hadamard-Cartan, sketch of proof
14 March 2011, 4.15pm Linearisation of the geodesic flow and Jacobi fields, Jacobi fields and comparison theorems

Last modified: 13.3.2011