# Riemannian Geometry Notion of a tensor. Metrics expressed in coordinates. Some readings: 1.

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Book reviews of books by Jost and Petersen, 2. Book review of book by Chavel. Some rather vague introductory notes to Differential Geometry an excerpt from my Thesis.

Tangent spaces and Riemannian manifolds

Petersen, Ch. HW 1a: Read Petersen, Ch. Verify the claim on p. Prove also the converse claim. September 11 Frame representation of metrics. Some more examples. Notion of a connection. HW 1b Some readings: 1. Chern, What is geometry? September 15, pm, Location: Krieger 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.

## Semi-Riemannian Geometry With Applications to Relativity, Volume 103

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?

## Riemannian Geometry

Show that this is well-defined i. 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.

### Submission history

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.

## Event: Riemannian Geometry and Generalised Functions | EMS

Also compute how the Ricci curvature changes under the rescaling of the metric. You should have also taken basic courses in linear algebra and real analysis. If you do not meet this criteria, but would still like to credit, please send me an email. Topics to be covered: Review of differentiable manifolds and tensors, Riemannian metrics, Levi-Civita connection, geodesics, exponential map, curvature tensor, first and second variation formulas, Jacobi fields, conjugate points and cut locus, Cartan-Hadamard and Bonnet Myers theorems.

Special topics if time permits - Comparison geometry theorems of Rauch, Toponogov, Bishop-Gromov , and Bochner technique. There will be 6 homeworks. The best five will be counted towards the grade.

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There is no late submission of homeworks. Riemannian Geometry Riemannian Geometry Riemannian Geometry Riemannian Geometry Riemannian Geometry Riemannian Geometry Riemannian Geometry Riemannian Geometry

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