Differential geometry of manifolds

In this text the authors have compiled all the basic topics and important concepts regarding differentiable Manifolds and different structures thereon. The primary aim is to cover the syllabi of different Indian Universities following the model U.G.C. curriculum. This book will also benefit researchers in the field of differential geometry and its applications in relativity theory, statistics, cosmology and biomathematics etc.

Some Preliminaries: Metric Spaces / Topological Spaces / Linear Algebra / Differentiable Manifolds: Definition and Examples of Differentiable Manifolds / Differentiable Functions on a Manifold / Tangent Spaces / Vector Fields on Differentiable Manifolds / Integral Curves and Flows / One Parameter Group of Transformations of a Manifold / Exterior Algebra and Exterior Derivative: Tensor Product / Tensor Algebra / Exterior Algebra / Exterior Derivative / Lie Group and Lie Algebras: Definition and Examples of Lie Groups / The Lie Algebra of a Lie Group / Lie Groups Homomorphism and Isomorphism / One Parameter Subgroup and Exponential Map / Lie Transformation Groups / Lei Derivative / Fibre Bundles: Principal Fibre Bundle / Linear Frame Bundle / Associated Principal Bundle / Vector Bundles / Bundle Homomorphism / Tangent Bundle / Linear Connections: Definition and Examples of Affine Connection / Torsion Tensor of an Affine Connection / Geodesic Normal Coordinates / Curvature Tensor of an Affine Connection / Riemannian Manifolds: Definition and Examples of Riemannian Manifolds / Riemannian Connection / Fundamental Theorem of Riemannian Geometry / Riemannian Curvature Tensor / Bianchi Identities / Ricci Identity / Ricci Tensor and Scalar Curvature / Geodesic of a Riemannian Manifold / Riemannian Manifold as a Metric Space / Gradient Vector Fields / Semi-symmetric Metric Connection / Sectional Curvatures / Schur's Theorem / Riemannian Manifold of Constant Curvature / Einstein Manifolds / Conformal, Concircular, Projective, Conharmonic Curvature Tensors / Locally Symmetric and semi-symmetric Riemannian Manifolds / Submanifolds: Submanifolds of a Riemannian Manifold / Induced Connection and Second Fundamental Form / Equations of Gauss-Codazzi and Ricci / Mean Curvature / Complex Manifold: Algebraic Preliminaries / Almost Complex Structure / Nijenhuis Tensor / Almost Hermite Manifold / Kähler Manifold / Almost Tachibana Manifold / Almost Product and Almost Decomposable Manifold / F-connection.

Undergraduate - Postgraduate Students in Mathematics