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Course11110889960 lectures

Riemannian Geometry

Prof. Ved Vivek Datar · IISc Bangalore

Start with lecture 1 At NPTEL

Lectures

Week 1 Introduction and Review of Prerequisites

  1. Angle Sum Property for Triangles Video
  2. Introduction to Curvature of Planar Curves Video
  3. Curvature of Embedded Surfaces in the 3-Dimensional Space Video
  4. Review of Smooth Manifolds Video
  5. Review of Tangent Spaces and Vector Fields Video

Week 2 Introduction to Riemannian Geometry

  1. Review of Cotangent Spaces, Smooth Forms and Tensors on Manifolds Video
  2. Introduction to Riemannian Manifolds Video
  3. Some More Examples of Riemannian Manifolds Video
  4. Isometries on Riemannian Manifolds Video
  5. The Flat Torus Video

Week 3 Covariant Differentiation of Vector Fields

  1. Local Orthonormal Frame and the Musical Isomorphisms Video
  2. Inner Products of Smooth Forms and the Riemannian Volume Element Video
  3. Introduction to Connections Video
  4. Continuing Further with the Discussion on Connections Video
  5. Covariant Differentiation along Smooth Curves Video

Week 4 Levi-Civita Connection and Riemann Curvature tensor

  1. Parallel Transport Map Video
  2. Levi-Civita Connection Video
  3. Covariant Differentiation of Tensors Video
  4. Riemann Curvature Tensor Video
  5. Properties of Curvature Tensor Video

Week 5 Sectional, Ricci and Scalar Curvature

  1. Curvature of Model Spaces Video
  2. Flat Riemannian Manifolds Video
  3. Sectional Curvature and Curvature Like Tensors Video
  4. Ricci Curvature Video
  5. Further Discussion on Ricci Curvature, and Scalar Curvature Video

Week 6 Geometry of Sub-manifolds, and Gauss-Bonnet formulae

  1. Geometry of Sub-manifolds Video
  2. Further Discussion on Geometry of Sub-manifolds Video
  3. Gauss-Bonnet formulae Video
  4. More Discussion on local Gauss-Bonnet formula Video
  5. Continuing Further with the Discussion on local Gauss-Bonnet formula Video

Week 7 Geodesics and First Variation Formula

  1. Geodesics Video
  2. Further Discussion on Geodesics Video
  3. Length and Energy Functionals, and the Variation Vector Field Video
  4. First Variation Formula Video
  5. Discussion on Critical Points of the Energy and Length Functionals Video

Week 8 Geodesics and the Exponential Map

  1. The Exponential Map Video
  2. Geodesic Normal Coordinates Video
  3. The Gauss Lemma Video
  4. Local Behaviour of Geodesics Video
  5. The Induced Metric Space Structure on a Riemannian Manifold Video

Week 9 Some More Topics Related to Geodesics: The Hopf-Rinow Theorem, The Second Variation Formula and Jacobi Fields

  1. The Hopf-Rinow Theorem Video
  2. The Hopf-Rinow Theorem (Contd.) Video
  3. The Second Variation Formula Video
  4. Jacobi Fields Video
  5. Jacobi Fields (Contd.) Video

Week 10 Further Study of Jacobi Fields and Introduction to the Concepts of Conjugate Points, Cut-Points, Segment…

  1. Taylor Expansion of the Metric in Terms of Geodesic Normal Coordinates Video
  2. Geometric Characterization of the Various Curvature Notions Video
  3. Conjugate Points and the Null Space of a Geodesic Video
  4. Obstructions for Geodesics to be Not Minimizing Video
  5. The Segment Domain and the Cut-Locus Video

Week 11 Cartan-Hadamard Theorem and Uniformization of Space Forms

  1. Fundamental Groups Video
  2. Covering Spaces Video
  3. Cartan-Hadamard Theorem Video
  4. Ambrose Proposition Video
  5. Uniformization of Space Forms Video

Week 12 Curvature and Topology

  1. Jacobi Fields in Space Forms Video
  2. Jacobi Fields Comparison Video
  3. Proof of Uniformization Theorem for Space Forms Video
  4. Myers Diameter Comparison Theorem Video
  5. Schur Theorem for Sectional Curvature and Ricci Curvature Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mathematics
Teacher
Prof. Ved Vivek Datar
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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