Riemannian Geometry

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

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Currently Playing: Lec 10 The Flat Torus

Flat torus, Lattice of points in the Euclidean space, Flat metric on torus

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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)