MATH4171

**Lecturer :** Norbert Peyerimhoff

**Terms :** Michaelmas 2010, Epiphany 2011

- 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**

- 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=- |

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