Riemannian Geometry Z7H339W5EB

URI	`m2://res/Z7H339W5EB`
Kind	course
Creators	Ved Vivek Datar (instructor)
Date	2025-12-23
MSC	53-XX
Scope	in
Ids	course 04dese585:111108899

Expressions

e1: edition

Language	en
Date	2025-12-23

m1: page, IISc Bangalore (record). nptel 111108899
m2: page, 2025-12-23 (record). nptel 111108899

Contents

part 1: Week 1 Introduction and Review of Prerequisites
session 1: Angle Sum Property for Triangles
session 2: Introduction to Curvature of Planar Curves
session 3: Curvature of Embedded Surfaces in the 3-Dimensional Space
session 4: Review of Smooth Manifolds
session 5: Review of Tangent Spaces and Vector Fields
part 2: Week 2 Introduction to Riemannian Geometry
session 6: Review of Cotangent Spaces, Smooth Forms and Tensors on Manifolds
session 7: Introduction to Riemannian Manifolds
session 8: Some More Examples of Riemannian Manifolds
session 9: Isometries on Riemannian Manifolds
session 10: The Flat Torus
part 3: Week 3 Covariant Differentiation of Vector Fields
session 11: Local Orthonormal Frame and the Musical Isomorphisms
session 12: Inner Products of Smooth Forms and the Riemannian Volume Element
session 13: Introduction to Connections
session 14: Continuing Further with the Discussion on Connections
session 15: Covariant Differentiation along Smooth Curves
part 4: Week 4 Levi-Civita Connection and Riemann Curvature tensor
session 16: Parallel Transport Map
session 17: Levi-Civita Connection
session 18: Covariant Differentiation of Tensors
session 19: Riemann Curvature Tensor
session 20: Properties of Curvature Tensor
part 5: Week 5 Sectional, Ricci and Scalar Curvature
session 21: Curvature of Model Spaces
session 22: Flat Riemannian Manifolds
session 23: Sectional Curvature and Curvature Like Tensors
session 24: Ricci Curvature
session 25: Further Discussion on Ricci Curvature, and Scalar Curvature
part 6: Week 6 Geometry of Sub-manifolds, and Gauss-Bonnet formulae
session 26: Geometry of Sub-manifolds
session 27: Further Discussion on Geometry of Sub-manifolds
session 28: Gauss-Bonnet formulae
session 29: More Discussion on local Gauss-Bonnet formula
session 30: Continuing Further with the Discussion on local Gauss-Bonnet formula
part 7: Week 7 Geodesics and First Variation Formula
session 31: Geodesics
session 32: Further Discussion on Geodesics
session 33: Length and Energy Functionals, and the Variation Vector Field
session 34: First Variation Formula
session 35: Discussion on Critical Points of the Energy and Length Functionals
part 8: Week 8 Geodesics and the Exponential Map
session 36: The Exponential Map
session 37: Geodesic Normal Coordinates
session 38: The Gauss Lemma
session 39: Local Behaviour of Geodesics
session 40: The Induced Metric Space Structure on a Riemannian Manifold
part 9: Week 9 Some More Topics Related to Geodesics: The Hopf-Rinow Theorem, The Second Variation Formula and Jacobi Fields
session 41: The Hopf-Rinow Theorem
session 42: The Hopf-Rinow Theorem (Contd.)
session 43: The Second Variation Formula
session 44: Jacobi Fields
session 45: Jacobi Fields (Contd.)
part 10: Week 10 Further Study of Jacobi Fields and Introduction to the Concepts of Conjugate Points, Cut-Points, Segment Domain and Cut-Locus
session 46: Taylor Expansion of the Metric in Terms of Geodesic Normal Coordinates
session 47: Geometric Characterization of the Various Curvature Notions
session 48: Conjugate Points and the Null Space of a Geodesic
session 49: Obstructions for Geodesics to be Not Minimizing
session 50: The Segment Domain and the Cut-Locus
part 11: Week 11 Cartan-Hadamard Theorem and Uniformization of Space Forms
session 51: Fundamental Groups
session 52: Covering Spaces
session 53: Cartan-Hadamard Theorem
session 54: Ambrose Proposition
session 55: Uniformization of Space Forms
part 12: Week 12 Curvature and Topology
session 56: Jacobi Fields in Space Forms
session 57: Jacobi Fields Comparison
session 58: Proof of Uniformization Theorem for Space Forms
session 59: Myers Diameter Comparison Theorem
session 60: Schur Theorem for Sectional Curvature and Ricci Curvature

Catalogue records

nptel: 111108899 (manifestation)
nptel_video: 111108899 (manifestation)