Differential Geometry of Manifolds – Vandana Prak

Differential Geometry of Manifolds is a mathematics textbook published by Vandana Prakashan, Gorakhpur. The book is authored by Prof. H.S. Shukla, Dr. B.N. Prasad and Prof. Dhruwa Narain Dubey. It covers important concepts of Riemannian Geometry and Differentiable Manifolds for students studying advanced mathematics.

Key Topics Covered

Section A: Riemannian Geometry

• Basic Concepts: Field, Vector Space, Vector Subspace, Linear Combination of Vectors, Linear Dependence and Independence, Basis and Dimension, Dual Space, Homomorphism, Differentiability Class, Topological Space, Connected Space and Compact Space.
• Tensor Algebra: Contravariant and Covariant Vectors, Tensor Products, Tensors of Different Types, Algebraic Operations, Contraction, Symmetric and Skew-Symmetric Tensors.
• Riemannian Geometry: Christoffel Symbols and Covariant Differentiation of Tensors.
• Geodesics and Parallelism of Vectors: Curvature of a Curve, Principal Normal, Geodesics, Geodesic Coordinates and Parallel Displacement of Vectors.
• Congruences and Orthogonal Ennuples: Congruences, Ricci Coefficients of Rotation, Geodesic Congruence and Normal Congruence.
• Curvature of a Riemannian Space: Curvature Tensor, Covariant Curvature Tensor, Bianchi Identity, Schur’s Theorem and Mean Curvature.
• Projective and Conformal Transformations: Geodesic Correspondence, Weyl’s Projective Curvature Tensor, Conformal Transformation and Conformal Curvature Tensor.

Section B: Differentiable Manifolds

• Differentiable Manifolds: Definitions, Examples, Differentiable Functions, Differentiable Mappings, Differentiable Curves, Tangent Space, Vector Field and Lie Bracket.
• Linear Connections: Invariant Viewpoint of Connections, Covariant Differential, Torsion and Curvature Tensors, Difference Tensor of Two Connections and Lie Derivatives.
• Differential Forms: Exterior Algebra, Interior Product or Contraction of a Form, Exterior Derivative and Cartan’s Structural Equations.
• Riemannian Manifold: Riemannian Metric, Riemannian Connection, Curvature Tensor and Sectional Curvature.
• Submanifolds and Hypersurface: Immersion, Normals, Induced Connection, Gauss Formula, Weingarten Formulae, Lines of Curvature, Mean Curvature and Equations of Gauss and Codazzi.

Book Details

• Book Title: Differential Geometry of Manifolds
• Publisher: Vandana Prakashan
• Authors: Prof. H.S. Shukla, Dr. B.N. Prasad and Prof. Dhruwa Narain Dubey
• Subject: Mathematics
• Main Sections: Riemannian Geometry and Differentiable Manifolds
• Language: English

This book is suitable for students looking for a structured reference on differential geometry, tensor algebra, Riemannian spaces and differentiable manifold theory. Please check the prescribed syllabus of your course before purchasing.