Lecture 10 of 60Week 2 Introduction to Riemannian Geometry
The Flat Torus
Prof. Ved Vivek Datar · IISc Bangalore
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- Riemannian Geometry, IISc Bangalore (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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All the lectures
- Angle Sum Property for Triangles
- Introduction to Curvature of Planar Curves
- Curvature of Embedded Surfaces in the 3-Dimensional Space
- Review of Smooth Manifolds
- Review of Tangent Spaces and Vector Fields
- Review of Cotangent Spaces, Smooth Forms and Tensors on Manifolds
- Introduction to Riemannian Manifolds
- Some More Examples of Riemannian Manifolds
- Isometries on Riemannian Manifolds
- The Flat Torus
- Local Orthonormal Frame and the Musical Isomorphisms
- Inner Products of Smooth Forms and the Riemannian Volume Element
- Introduction to Connections
- Continuing Further with the Discussion on Connections
- Covariant Differentiation along Smooth Curves
- Parallel Transport Map
- Levi-Civita Connection
- Covariant Differentiation of Tensors
- Riemann Curvature Tensor
- Properties of Curvature Tensor
- Curvature of Model Spaces
- Flat Riemannian Manifolds
- Sectional Curvature and Curvature Like Tensors
- Ricci Curvature
- Further Discussion on Ricci Curvature, and Scalar Curvature
- Geometry of Sub-manifolds
- Further Discussion on Geometry of Sub-manifolds
- Gauss-Bonnet formulae
- More Discussion on local Gauss-Bonnet formula
- Continuing Further with the Discussion on local Gauss-Bonnet formula
- Geodesics
- Further Discussion on Geodesics
- Length and Energy Functionals, and the Variation Vector Field
- First Variation Formula
- Discussion on Critical Points of the Energy and Length Functionals
- The Exponential Map
- Geodesic Normal Coordinates
- The Gauss Lemma
- Local Behaviour of Geodesics
- The Induced Metric Space Structure on a Riemannian Manifold
- The Hopf-Rinow Theorem
- The Hopf-Rinow Theorem (Contd.)
- The Second Variation Formula
- Jacobi Fields
- Jacobi Fields (Contd.)
- Taylor Expansion of the Metric in Terms of Geodesic Normal Coordinates
- Geometric Characterization of the Various Curvature Notions
- Conjugate Points and the Null Space of a Geodesic
- Obstructions for Geodesics to be Not Minimizing
- The Segment Domain and the Cut-Locus
- Fundamental Groups
- Covering Spaces
- Cartan-Hadamard Theorem
- Ambrose Proposition
- Uniformization of Space Forms
- Jacobi Fields in Space Forms
- Jacobi Fields Comparison
- Proof of Uniformization Theorem for Space Forms
- Myers Diameter Comparison Theorem
- Schur Theorem for Sectional Curvature and Ricci Curvature