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Indian Institute of Science Education and Research Bhopal


MTH 613: Introduction to Riemannian Geometry (4)

Pre-requisites: MTH 405 and MTH 508

Review of differentiable manifolds: vector bundles, tensors, vector fields, differential forms, Lie groups

Riemannian metrics. Definition, examples, existence theorem; model spaces of Riemannian geometry

Connections: connections on a vector bundle, linear connections, covariant derivative, parallel transport,  geodesics

Riemannian connections and geodesics: torsion tensor, Fundamental Theorem of Riemannian Geometry, geodesics of the model spaces, exponential map,  convex neighborhoods, Riemannian distance function, first variation formula,  Gauss' lemma, geodesics as locally minimizing curves; completeness, statement of Hopf-Rinow Theorem

Curvature: Riemann Curvature Tensor, Bianchi identity, scalar, sectional and Ricci curvatures  

Jacobi Fields: Jacobi equation, conjugate points, second variation formula, spaces of constant curvature (if time permits)  

Curvature and topology: Gauss-Bonnet Theorem, Bonnet-Myers Theorem, Cartan-Hadamard Theorem

Suggested Books:



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