NOC:Riemannian Geometry

Lecture 1 - Angle Sum Property for Triangles

Lecture 2 - Introduction to Curvature of Planar Curves

Lecture 3 - Curvature of Embedded Surfaces in the 3-Dimensional Space

Lecture 4 - Review of Smooth Manifolds

Lecture 5 - Review of Tangent Spaces and Vector Fields

Lecture 6 - Review of Cotangent Spaces, Smooth Forms and Tensors on Manifolds

Lecture 7 - Introduction to Riemannian Manifolds

Lecture 8 - Some More Examples of Riemannian Manifolds

Lecture 9 - Isometries on Riemannian Manifolds

Lecture 10 - The Flat Torus

Lecture 11 - Local Orthonormal Frame and the Musical Isomorphisms

Lecture 12 - Inner Products of Smooth Forms and the Riemannian Volume Element

Lecture 13 - Introduction to Connections

Lecture 14 - Continuing Further with the Discussion on Connections

Lecture 15 - Covariant Differentiation along Smooth Curves

Lecture 16 - Parallel Transport Map

Lecture 17 - Levi-Civita Connection

Lecture 18 - Covariant Differentiation of Tensors

Lecture 19 - Riemann Curvature Tensor

Lecture 20 - Properties of Curvature Tensor

Lecture 21 - Curvature of Model Spaces

Lecture 22 - Flat Riemannian Manifolds

Lecture 23 - Sectional Curvature and Curvature Like Tensors

Lecture 24 - Ricci Curvature

Lecture 25 - Further Discussion on Ricci Curvature, and Scalar Curvature

Lecture 26 - Geometry of Sub-manifolds

Lecture 27 - Further Discussion on Geometry of Sub-manifolds

Lecture 28 - Gauss-Bonnet formulae

Lecture 29 - More Discussion on local Gauss-Bonnet formula

Lecture 30 - Continuing Further with the Discussion on local Gauss-Bonnet formula

Lecture 31 - Geodesics

Lecture 32 - Further Discussion on Geodesics

Lecture 33 - Length and Energy Functionals, and the Variation Vector Field

Lecture 34 - First Variation Formula

Lecture 35 - Discussion on Critical Points of the Energy and Length Functionals

Lecture 36 - The Exponential Map

Lecture 37 - Geodesic Normal Coordinates

Lecture 38 - The Gauss Lemma

Lecture 39 - Local Behaviour of Geodesics

Lecture 40 - The Induced Metric Space Structure on a Riemannian Manifold

Lecture 41 - The Hopf-Rinow Theorem

Lecture 42 - The Hopf-Rinow Theorem (Continued...)

Lecture 43 - The Second Variation Formula

Lecture 44 - Jacobi Fields

Lecture 45 - Jacobi Fields (Continued...)

Lecture 46 - Taylor Expansion of the Metric in Terms of Geodesic Normal Coordinates

Lecture 47 - Geometric Characterization of the Various Curvature Notions

Lecture 48 - Conjugate Points and the Null Space of a Geodesic

Lecture 49 - Obstructions for Geodesics to be Not Minimizing

Lecture 50 - The Segment Domain and the Cut-Locus

Lecture 51 - Fundamental Groups

Lecture 52 - Covering Spaces

Lecture 53 - Cartan-Hadamard Theorem

Lecture 54 - Ambrose Proposition

Lecture 55 - Uniformization of Space Forms

Lecture 56 - Jacobi Fields in Space Forms

Lecture 57 - Jacobi Fields Comparison

Lecture 58 - Proof of Uniformization Theorem for Space Forms

Lecture 59 - Myers Diameter Comparison Theorem

Lecture 60 - Schur Theorem for Sectional Curvature and Ricci Curvature