Riemannian Geometry

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Curated by: NPTEL - Indian Institute of Science, Bengaluru (61 videos)

Currently Playing: Lec 43 The Second Variation Formula

Second derivative test, Second variation formula, Admissible paths, One-parameter admissible variation, Admissible vector fields, Variation vector field, Proper variation, Energy functional, Index form, Length minimizing geodesic/Minimal geodesic

Tracks in this Playlist

Lec 60 Schur Theorem for Sectional Curvature and Ricci Curvature
Lec 59 Myers Diameter Comparison Theorem
Lec 58 Proof of Uniformization Theorem for Space Forms
Lec 57 Jacobi Fields Comparison
Lec 56 Jacobi Fields in Space Forms
Lec 55 Uniformization of Space Forms
Lec 54 Ambrose Proposition
Lec 53 Cartan-Hadamard Theorem
Lec 52 Covering Spaces
Lec 51 Fundamental Groups
Lec 50 The Segment Domain and the Cut-Locus
Lec 49 Obstructions for Geodesics to be Not Minimizing
Lec 48 Conjugate Points and the Null Space of a Geodesic
Lec 47 Geometric Characterization of the Various Curvature Notions
Lec 46 Taylor Expansion of the Metric in Terms of Geodesic Normal Coordinates
Lec 45 Jacobi Fields (Contd.)
Lec 44 Jacobi Fields
Lec 43 The Second Variation Formula
Lec 42 The Hopf-Rinow Theorem (Contd.)
Lec 41 The Hopf-Rinow Theorem
Lec 40 The Induced Metric Space Structure on a Riemannian Manifold
Lec 39 Local Behaviour of Geodesics
Lec 38 The Gauss Lemma
Lec 37 Geodesic Normal Coordinates
Lec 36 The Exponential Map
Lec 35 Discussion on Critical Points of the Energy and Length Functionals
Lec 34 First Variation Formula
Lec 33 Length and Energy Functionals, and the Variation Vector Field
Lec 32 Further Discussion on Geodesics
Lec 31 Geodesics
Lec 30 Continuing Further with the Discussion on local Gauss-Bonnet formula
Lec 29 More Discussion on local Gauss-Bonnet formula
Lec 28 Gauss-Bonnet formulae
Lec 27 Further Discussion on Geometry of Sub-manifolds
Lec 26 Geometry of Sub-manifolds
Lec 25 Further Discussion on Ricci Curvature, and Scalar Curvature
Lec 24 Ricci Curvature
Lec 23 Sectional Curvature and Curvature Like Tensors
Lec 22 Flat Riemannian Manifolds
Lec 21 Curvature of Model Spaces
Lec 20 Properties of Curvature Tensor
Lec 19 Riemann Curvature Tensor
Lec 18 Covariant Differentiation of Tensors
Lec 17 Levi-Civita Connection
Lec 16 Parallel Transport Map
Lec 15 Covariant Differentiation along Smooth Curves
Lec 14 Continuing Further with the Discussion on Connections
Lec 13 Introduction to Connections
Lec 12 Inner Products of Smooth Forms and the Riemannian Volume Element
Lec 11 Local Orthonormal Frame and the Musical Isomorphisms
Lec 10 The Flat Torus
Lec 09 Isometries on Riemannian Manifolds
Lec 08 Some More Examples of Riemannian Manifolds
Lec 07 Introduction to Riemannian Manifolds
Lec 06 Review of Cotangent Spaces, Smooth Forms and Tensors on Manifolds
Lec 05 Review of Tangent Spaces and Vector Fields
Lec 04 Review of Smooth Manifolds
Lec 03 Curvature of Embedded Surfaces in the 3-Dimensional Space
Lec 02 Introduction to Curvature of Planar Curves
Lec 01 Angle Sum Property for Triangles
Riemannian Geometry (Intro)