MA 109a Introduction to Geometry and geometric analysis (Course at Caltech 1995)

This is the first part of an introduction into Geometry and geometric analysis. This course was given in the first term 1995 at Caltech for undergraduate students.
The (1 MBytes PS-file) of the course is still available but has not been revised since the material was handed out to the participants of the course. A highlight of the course is an introduction into general relativity for mathematicians. The course contains about 60 exercices with solutions for the 10 week course.
To the content:
  1. Chapter 1: Manifolds (definition, examples, diffeomorphisms, constructing manifolds, theorem of Sard partition of Unity, Whitney's embedding theorem, Brower's fixed point theorem)
  2. Chapter 2: Tensor analysis (general tensors, antisymmetric tensors, tangent space and tensor fields, exterior derivative, integration on manifolds, chains and boundaries, theorem of Stokes for chains, Theorem of Stokes for oritented manifolds)
  3. Chapter 3: Riemannian geometry (metric tensors, Hodge star operation, Riemannian manifolds, Theorem of Stokes for Riemannian manifolds, connections, covariant derivative, parallel transport, geodesics, Riemannian curvature, Ricci tensor and scalar curvature, the second Bianchi identity, General relativity).

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