Math 7210 – Riemannian Geometry
In Spring semester 2017, I am teaching Math 7210: Riemannian Geometry.
Office hours:
MWF 11:00am–12:00pm and by appointment
Information:
Term: Spring 2017 Time: MWF 10:10–11:00am Location: 1312 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 Blackboard 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.
Mon Jan 9 — Discussion of syllabus, §0.1 Introduction; Calculus review; §0.2: Differentiable manifolds Wed Jan 11 — §0.2: Differentiable manifolds, smooth mappings, tangent spaces Fri Jan 13 — §0.2–0.4: Derivations in \(\mathbb{R}^n\); bases for tangent space; derivative of smooth map; tangent bundle Wed Jan 18 — §0.3–0.5: Diffeomorphisms; immersions & embeddings; submersion theorem; vector fields Fri Jan 20 — §0.5–1.2: bracket of vector fields; local flow of a vector field; Riemannian metrics Mon Jan 23 — §1.2: Riemannian metrics & manifolds; Examples; Lie groups; Adjoint representation Wed Jan 25 — §1.2: Lengths of curves; Tensor & Exterior product; multilinear maps; alternating maps Fri Jan 27 — §1.2: Cotangent bundle; Tensor fields; Volume forms & Riemannian volume; Integration Mon Jan 30 — §2.1–2.2: Intuitive idea of covariant derivative; Affine connections Wed Feb 1 — §2.2–2.3: Covariant derivative; Parallelism; Parallel transport; Compatible connections Fri Feb 3 — §2.3: Proof of existence/uniquenesss of the (Levi-Civita) Riemannian connection Mon Feb 6 — §3.1–3.2: Example of covariant derivative; Definition of geodesc; Geodesic field; Geodesic flow Wed Feb 8 — §3.2-3.3: Homogeneity for geodesic flow; Exponential map; Paramerized surfaces & Vector fields Fri Feb 10 — §3.3: Gauss's Lemma; Geodesics locally minimize length Mon Feb 13 — §3.3–4.1: Length minimzing paths are geodeiscs; Examples; Convexity; Gaussian Curvature Wed Feb 15 — §4.2: Intuition of Sectional Curvature; Definition of the Curvature Tensor and its basic properties Fri Feb 17 — §4.2: Curvature depends only on values at a point; (0,4) curvature tensor; Sectional curvature. Mon Feb 20 — §4.3–4.4: Sectional curvature determines full curvature; Ricci curvature; Scalar curvature. Wed Feb 22 — §4.4: Interpretations of curvature: parallel transport and horizontal vector vields on \(TM\). Fri Feb 24 — §4.4–5.2: Covariant derivative of tensors; Definition of Jacobi fields. Mon Feb 27 — §5.2: Solving Jacobi equation; characterize Jacobi fields \(J(t)\) with \(J(0)=0\) Wed Mar 1 — §5.2–5.3: Estimate the norm \(\vert J(t)\vert\) of Jacobi fields; Conjugate points; Conjugate locus. Fri Mar 3 — §5.3–6.1: Critical pts of exponential map; Pairing Jacobi and velocity fields; Isometric immersions Mon Mar 13 — §6.2: The Second Fundamental Form and its associated self-adjoint operator \(S_\eta\) of \(T_p M\) Wed Mar 15 — §6.2: SFF in codimension 1: Gauss spherical mapping; Relation to sectional curvature Fri Mar 17 — §6.2: Minimal & Totally Geodesic immersions; Gaussian vs Sectional curvature; Curv. of Sphere Mon Mar 20 — §6.3: Normal connection; Normal curvature tensor; Gauss Equation; Ricci Equation Wed Mar 22 — §6.3–7.2: Codazzi Equation; Complete Riemannian manifolds; Distance function / metric Fri Mar 24 — §7.2: The Hopf-Rinow Theorem about complete Riemannian manifolds Mon Mar 27 — §7.3: The Hadamard Theorem Wed Mar 29 — §8.1–8.2: Spaces of constant curvature; Cartan's theorem determining metric from curvature Fri Mar 31 — §8.3: Hyperbolic space: the Upper Half-Space and Hyperboloid models Mon Apr 3 — §8.3–8.4: Hyperboloid model (geodesics & curvature); Classification of space forms Wed Apr 5 — §8.4: Classification of space forms; Mostow Rigidity; non-rigidity in dimension 2 Fri Apr 7 — §9.1–9.3: Student Presentation: Variations of energy and the Bonnet-Myers Theorem Mon Apr 10 — Student Presentation: de Rham cohomology & the Hodge Decomposition Theorem Wed Apr 12 — Student Presentation: The 2-dimensional Gauss–Bonnet theorem Fri Apr 14 — Student Presentation: Symmetric spaces Mon Apr 17 — Student Presentations: Symmetric spaces & the 2-dimensional Gauss–Bonnet theorem Wed Apr 19 — Student Presentation: Chern classes Fri Apr 21 — Student Presentation: Morse theory Mon Apr 24 — Student Presentations: Chern classes & Morse theory