Math 7210 – Riemannian Geometry
Office hours:
Mon 3:00pm–4:00pm Tues 1:00pm–3:00pm Fri 11:00am–12:00pm and by appointment
Information:
Term: Fall 2019 Time: MWF 1:10–2:00pm Location: 1313 Stevenson Center
Links:
View the Course Syllabus for the course description and course policies. View the Assignments Page for the list of homework assignments. Access our Brightspace page (login required) to find your Grades for the class.
Announcements:
- There are currently no announcements.
Schedule:
Here is a tentative and ever-evolving schedule of the topics covered in each class. I will aim to have the topics for each class posted at least a class period in advance, and will modify the schedule after the fact to reflect what was actually covered.
Wed Aug 21 — Discussion of syllabus, §0.1 Introduction; Calculus review; §0.2: Differentiable manifolds Fri Aug 23 — §0.2: Smooth mappings; Tangent spaces; Derivations Mon Aug 26 — §0.2: Derivatives; Diffeomorphisms; §0.3 Immersions & Embeddings; Submersion Theorem Wed Aug 28 — §0.4: Quotients; Orientations; Tangent bundles; §0.5 Vector fields. Fri Aug 30 — §0.5–1.2: bracket of vector fields; local flow of a vector field; Riemannian metrics Mon Sept 2 — §1.2: Riemannian metrics & manifolds; Examples; Lie groups; Adjoint representation Wed Sept 4 — §1.2: Lie groups: left-invariant vector fields and flows, exponential map. Fri Sept 6 — §1.2: Lengths of curves; Tensor & Exterior product; multilinear maps; alternating maps Mon Sept 9 — §1.2: Tensors; Differential forms; Volume forms; Integration §2.2 Affine connections Wed Sept 11 — No lecture; Students can meet to discuss recent material, assignments, additonal exercises. Fri Sept 13 — §2.1–2.2: Intuitive idea of covariant derivative; Affine connections; Parallelism Mon Sept 16 — §2.2–2.3: Parallel transport; Compatible, Symmetric, and Riemannian connections Wed Sept 18 — §2.3: Existence and uniqueness of Levi-Civita (Riemannian) connection; E.g. hyperbolic plane Fri Sept 20 — §3.1–3.2: Intuitive notion of geodesics; Geodesic flow & existence of geodesics; Homogeneity Mon Sept 23 — §3.2: The Exponential Map; Parameterized surfaces; Gauss' Lemma Wed Sept 25 — §3.2–3.3: Proof of Gauss' Lemma; length-minimizing properties of geodesics Fri Sept 27 — §3.3: Length minimizing paths are geodesic; §3.4: Convex neighborhoods; §4.1: Curvature intro Mon Sept 30 — §4.2: Curvature – Definition, properties, Bianchi identity Wed Oct 2 — §4.2: Properties of curvature; &secdt;4.3: Sectional Curvature Fri Oct 4 — §4.3: Constant sectional curvatures; §4.4: Ricci & Scalar Curvatures Mon Oct 7 — Curvature of Parametized surfaces; Curvature and path-dependence of parallel transport. Wed Oct 9 — Sudent presentations of Exercises from Homework 5. Fri Oct 11 — Student presentations; Interpretation of curvature via horizontal distribution on tangent bundle. Mon Oct 14 — §5.1–5.2: Jacobi fields and the Jacobi Equation. Wed Oct 16 — §5.2–5.3: Estimate the norm \(\vert J(t)\vert\) of Jacobi fields; Conjugate points; Conjugate locus. Fri Oct 18 — §5.3–6.1: Critical pts of exponential map; Pairing Jacobi and velocity fields; Isometric immersions. Mon Oct 21 — §6.2: The Second Fundamental Form and its associated self-adjoint operator \(S_\eta\) of \(T_p M\). Wed Oct 23 — §6.2: SFF in codimension 1: Gauss spherical mapping; Relation to sectional curvature. Fri Oct 25 — No class (fall break). Mon Oct 28 — §6.2: Gaussian vs Sectional curvature; Curv. of Sphere; Totally Geodesic immersions. Wed Oct 30 — §6.2: Minimal immersions; §6.1–6.2: Complete manifolds; §8.1: Spaces of const curvature. Fri Nov 1 — §8.1–8.2: Spaces of constant curvature; Cartan's theorem determining metric from curvature. Mon Nov 4 — §12.1–12.3: Preisman's Theorem on fundamental group of manifolds with negative curvature. Wed Nov 6 — Student Presentation (Sumati): Hopf-Rinow Theorem. Fri Nov 8 — Student Presentation (Minh): Hadamard Theorem. Mon Nov 11 — Student Presentation (Jiawei): Bonnet–Myers Theorem. Wed Nov 13 — Student Presentation (Sifan): Tensor Calculus and Lorenzian geometry. Fri Nov 15 — No leture. Mon Nov 18 — Student Presentation (David): Vector Bundles and Swan's Theorem. Wed Nov 20 — Student Presentation (Shaoyang): Hyperbolic Space: Isometries and Liouville Theorem. Fri Nov 22 — Student Presentation (Dylan): Symplectic Manifolds and Hamiltonian mechanics. Mon Dec 2 — Student Presentation (Sam): Kähler Manifolds.