## Riemannian Geometry (course 110.645)

### Date: Tuesdays, Thursdays at 11:45am. Room: Krieger Hall 204.

Teacher: Y.A. Rubinstein.

Course plan:
The course will be divided roughly into three parts. In the first part we will discuss the fundamental notions of connection and curvature, the geometry of submanifolds, metric properties of geodesics and Jacobi theory. In the second part we will review some of the results of the first part from the point of view of distance functions and polar coordinates and also compute explicitly some concrete examples. We will also discuss the Gauss-Bonnet theorem. In the third part we will discuss some of the basic results on sectional and Ricci curvature comparison, the maximum principle and a brief introduction to convergence theory of Riemannian manifolds.

Main reference:
P. Petersen, Riemannian geometry (2nd Ed.).

J. Jost, Riemannian geometry and geometric analysis (5th Ed.).
M.M. Postnikov, Geometry VI: Riemannian geometry.

Assignments:
Homework on a regular basis and a 25-minute presentation at the end of the course. Homework from previous week due on the next Tuesday.

Schedule:

• September 9
Overview/syllabus/references. Structures on manifolds. Examples of Riemannian metrics. Notion of a tensor. Metrics expressed in coordinates.

1. Book reviews of books by Jost and Petersen,
2. Book review of book by Chavel.
3. Some rather vague introductory notes to Differential Geometry (an excerpt from my Thesis).
4. Petersen, Ch. 1: Sections 1,3, Ch. 2: Sections 1,2,3,6.

HW 1a: Read Petersen, Ch. 1, Sec. 3.4, p. 12-13. Verify the claim on p. 13 that all odd-order derivatives of $\psi$ at $0$ must vanish in order for the surface to be smooth at $0$. Prove also the converse claim.
• September 11
Frame representation of metrics. Some more examples. Notion of a connection.

HW 1b

1. S.-S. Chern, What is geometry?

• September 15, 4:30pm, Location: Krieger 302 (note special time and location)
Notion of a connection. The Levi-Civita connection and some of its properties. Derivatives of tensors. Notions of Hessian, Laplacian, divergence, second covariant derivative.
• September 16
Notion of curvature. Properties of the Riemann curvature tensor. The Ricci, scalar and sectional curvatures.

HW 2: 1. Prove that the sectional curvatures completely determine the Riemann curvature tensor. Can you compute (using the symmetries of this tensor) the number of independent sectional curvatures?
2. The volume form of an $n$-dimensional Riemannian manifold is given pointwise by the wedge product of $n$ 1-forms that form an oriented orthonormal basis for the cotangent bundle. Show that this is well-defined (i.e., independent of such a choice of a basis). Then show that in local coordinates it is given by $f dx$ where $f$ is the square root of the determinant of the metric tensor expressed in those coordinates, and $dx$ is the wedge prodcut of $dx^1, dx^2, ..., dx^n$.
• September 23
Frames normal at a point. The symmetries of the curvature tensor.

HW 3
• September 25
Count of the number of independent curvature coefficients of the curvature tensor. The curvature operator. Curvature tensor characterization of constant sectional curvature. Introduction to distance functions. The fundamental curvature equations: tangential and normal curvature equations.
• September 30
The curvature of spheres. The determinant of the shape operator. The fundamental curvature equations: radial curvature equation. The curvature of surfaces.

HW 4: 1. Petersen, p. 62, exercise 28. Also compute how the Ricci curvature changes under the rescaling of the metric.
2. Petersen, p. 62, exercise 31 but you can ignore the last part of the exercise, about the equivalence of the closedness of the 2-form and the paralleness of the complex structure.
3. Familiarize yourself with Petersen, pp. 375-383 (sections 1-3 of the Appendix).
• October 2
Curvature computations: product of spheres, rotationally symmetric metrics.
• October 7
Curvature of rotationally symmetric metrics.

HW 5
• October 14
The Lie group SU(2), the Berger spheres and their curvature. The Hopf fibration.

HW 6
• October 16
Spheres as warped products. Joins of topological spaces. The Hopf fibration continued.
• October 20, 4:30pm, Location: Krieger 204 (note special time)
Complex projective space and the generalized Hopf fibration.
• October 21
Curvature of the complex projective plane.

HW 7
• November 4
Basics of hypersurface theory. The Berger spheres and the complex projective plane are not hypersurfaces.
• November 6
Parallel transport. Geodesics. Parallel transport on the 2-sphere.

HW 8
• November 11
The round 3-sphere is isometric to SU(2) with the biinvarint metric. The metric space structure of a Riemannian manifold.
• November 13
Segments. First variation for geodesics. Characterization of segments.

HW 9
• November 18
Jacobi theory: Jacobi fields as critical points for the Lagrangian coming from the second variation of energy.
• November 20
Jacobi theory: minimal fields, Jacobi variations.
• November 25
Jacobi fields: the vector space of Jacobi fields, normal Jacobi fields and the Jacobi theorem on the characterization of minimizing geodeiscs via conjugate points.
• December 2
The Gauss Lemma and some consequences.
• December 4
Normal and geodesic coordinates. On the characterization of the cut-locus.

HW 10
• December 9, 10am, Krieger 413 (note the special time and location)
Student presentations.