Two different topics are offered, students participate actively in one of the topics.
Topic 1: Measure and Category
This topic uses the
book Measure and Category by John C. Oxtoby.
Topic 2: Differential manifolds and Differential geometry
Talk 1: Introduction to smooth manifolds
Literature: [5]
- Definition of topological manifolds, charts and atlases (Chapter 1, pp.1-5)
- Examples: spheres, projective spaces, product manifolds, tori (Examples 1.2-1.5)
- Optional: Topological properties of manifolds (pp. 8-11)
- Smooth manifolds (pp. 11-17)
- Examples: Euclidean space, different smooth structure on the real line, spheres, projective spaces, products (Examples 1.13, 1.14, 1.20, 1.21, 1.22)
Talk 2: Smooth maps
Literature: [5]
- Definition and properties of smooth functions and smooth maps on a manifold (Chapter 2, pp. 30-35)
- Examples: inclusions of spheres into Euclidean space, quotient maps from Euclidean space without the origin to projective space, and from a sphere to projective space (Examples 2.5)
- Optional: Definition of diffeomorphisms (p. 36-37)
- Smooth partitions of unity (pp. 49-57)
Talk 3: Vector bundles and the tangent bundle
Literature: [5]
- Definition of a vector bundle (Chapter 5, pp. 103-105)
- Examples: product bundles, M??bius bundle (Examples 5.1, 5.2)
- Transition functions and constructions of vector bundles (pp. 107-109)
- Tangent vectors on smooth manifolds (Chapter 3, pp. 60-65)
- The tangent bundle (Chapter 4, pp. 80-82 and Proposition 5.3)
Talk 4: Derivatives of maps
Literature: [5]
- Bundle maps (Chapter 5, pp. 115-118)
- Pushforwards (or maps of tangent spaces) (Chapter 3, pp. 65-68)
- Computations (pp. 69-73)
- Tangent vectors to curves (pp. 75-77)
*Talk 5: The Whitney Embedding Theorem
Literature: [5]
- Whitney's immersion and embedding theorems (Chapter 10, pp. 241-251)
- Optional: Whitney's approximation theorem (pp. 252-259)
Talk 6: Tensors and differential forms
Literature: [5]
- Recollection: tensors and tensor products of vector spaces (Chapter 11, pp. 260-268)
- Tensor fields on manifolds (pp. 269-271)
- Alternating tensors (Chapter 12, pp. 291-299)
- The wedge product (pp. 299-302)
- Differential forms on manifolds and the exterior derivative (pp. 302-313)
Talk 7: Integration on manifolds and Stokes' Theorem
Literature: [5]
- Oriented and orientable manifolds (Chapter 13, pp. 324-329)
- Integrals of differential forms on manifolds (Chapter 14, pp. 349-358)
- Stokes' Theorem with proof (pp. 359-362)
Talk 8: Riemannian metrics and Riemannian manifolds
Literature: [6]
- Definition and examples of Riemannian metrics on smooth manifolds (Chapter 3, pp. 23-26)
- The volume element (pp. 29-30)
- Definition of a connection on a smooth manifold (Chapter 4, pp. 47-51)
- Covariant derivatives (pp. 53-55)
- Vector fields and derivatives along curves (Chapter 4, pp. 55-58)
Talk 9: Geodesics and curve lengths
Literature: [6]
- Definition of geodesics; local existence and uniqueness (pp. 58-59)
- Parallel transport (pp. 59-62)
- The Levi-Civita connection and its existence and uniqueness (Chapter 5, pp. 65-71)
- Curve length (Chapter 6, pp. 91-94)
- From the Chapter Geodesics and Minimizing Curves (pp. 96-107): results and as many proofs as possible within the time
Talk 10: Euclidean, hyperbolic, and spherical n-space and their geometries
Literature: [6]
- The model spaces as Riemannian manifolds (Chapter 3, pp. 33-42)
- Geodesics in the model spaces (Chapter 5, pp. 81-86)
- Surfaces with constant curvature (I need to find a good reference)
- Optional: Geodesic triangles in the model spaces
Talk 11: Curvature
Literature: [6]
- Definition of the curvature tensor (Chapter 7, pp. 115-119)
- Properties of the curvature tensor, Bianchi identities (pp. 121-124)
- curvature zero means locally isometric with Euclidean space (pp. 119-121)
- Optional: other notions of curvature (pp. 124-127)
- Examples
Talk 12: Submanifolds, Gaussian curvature, and the Theorema Egregium
Literature: [6]
- Riemannian submanifolds and the second fundamental form (Chapter 8, pp. 131-137)
- Application: curvature of curves (pp. 137-139)
- Hypersurfaces of Euclidean space and Gaussian curvature (pp. 139-143)
- Optional: The Gauss map (Problem 8-6)
- Gauss's Theorema Egregium with proof (pp. 143-145)
Talk 13: The Gauss-Bonnet theorem
Literature: [6]
- Preparatory plane geometry (Chapter 9, pp. 155-162)
- The Gauss-Bonnet formula (pp. 162-166)
- Triangulations of surfaces and the Gauss-Bonnet theorem (pp. 166-170)
- Optional: Characterization of the Euler characteristic by de Rham cohomology (I need to find literature)
*Talk 14: Bonnet's theorem
Literature: [6]
- Complete manifolds and the Hopf-Rinow theorem (Chapter 6, pp. 108-111)
- Bonnet's and Myer's theorems (Chapter 11, pp. 200-202 plus relevant parts from Chapter 10)
- Optional: Classification of simply-connected manifolds of constant curvature. (pp. 204-207)
Literature list
Above I have only referred to [5] and [6], but looking at the relevant chapters in other books will help you prepare your talk.
[1] Berger, Gostiaux: Differential geometry: manifolds, curves, and surfaces
[2] Bredon: Topology and geometry
[3] do Carmo: Riemannian geometry
[4] Gallot, Hulin, Lafontaine: Riemannian geometry
[5] Lee: Introduction to smooth manifolds
[6] Lee: Riemannian manifolds
