Kindly note that we have put a lot of effort into researching the best books on Differential Geometry of Manifolds subject and came out with a recommended list of top 10 best books. The table below contains the Name of these best books, their authors, publishers and an unbiased review of books on "Differential Geometry of Manifolds" as well as links to the Amazon website to directly purchase these books. As an Amazon Associate, we earn from qualifying purchases, but this does not impact our reviews, comparisons, and listing of these top books; the table serves as a ready reckoner list of these best books.
1. “A Course in Tensors with Applications to Riemannian Geometry” by R S Mishra
“A Course in Tensors with Applications to Riemannian Geometry” Book Review: This book offers a systematic study of tensors and Riemannian geometry. The book derives the laws of transformations of contravariant vectors while also dealing with exterior algebra, tensor algebra, tensor calculus, differentiable manifolds, and tangent vector spaces. More advanced topics are also covered with the help of examples and exercises. Basic understanding of groups, rings, fields, and vector spaces is necessary. Students, professors, researchers, and professionals can refer to this book.


2. “Differentiable Manifolds” by Y Matsushima
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“Differentiable Manifolds” Book Review: This book introduces the theory of differential manifolds and Lie groups in a comprehensive and lucid manner. Prerequisite knowledge of point set topology, basic analysis, vector spaces, groups, and other elements of algebra. The book can be used for an advanced undergraduate course or an introductory graduate course.


3. “Differential Geometry with Applications to Mechanics and Physics” by Y Talpiert
“Differential Geometry with Applications to Mechanics and Physics” Book Review: This book covers differential geometry along with its applications to mechanics and physics. The book starts with an introduction to topology and differential calculus in banach spaces, differentiable manifolds and mapping submanifolds, and tangent vector space. This is followed by an indepth discussion on tangent bundle, vector field on manifold, Lie algebra structure, and oneparameter group of diffeomorphisms. Subsequent chapters deal with exterior differential forms, Lie derivative and Lie algebra, nform integration on nmanifold, and Riemann geometry. It is replete with examples and solved exercises.


4. “Differential Geometry of Manifolds” by Khan
“Differential Geometry of Manifolds” Book Review: This book presents a comprehensive coverage of the theory of differential and Riemannian manifolds by adopting a coordinatefree approach. Fundamentals of bundles, exterior algebra and calculus, Lie group and its algebra and calculus are provided in great detail. The book also discusses Riemannian geometry, submanifolds and hypersurfaces, almost complex manifolds, etc. in a lucid manner. Numerous illustrations, solved examples, and exercises are also provided. The book is designed for the postgraduate students of mathematics as well as researchers working in differential geometry and its applications to general theory of relativity and cosmology, and other applied areas.
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5. “Differential Geometry Of Manifolds” by U C De and A A Shaiks
“Differential Geometry of Manifolds” Book Review: This book presents a detailed treatment of the theory of differentiable and Riemannian manifolds. The book discusses the tangent vector’s importance in the study of differentiable manifolds. Numerous proofs are provided to explain the theory of Riemannian geometry in detail. This book is ideal for postgraduate students and researchers working in differential geometry and its applications to general relativity and cosmology.


6. “Differential Geometry of Manifolds” by Stephen T Lovett
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“Differential Geometry of Manifolds” Book Review: This book expands the fundamental concepts of manifolds from curves and surfaces of differential geometry. An introduction to the Hamiltonian formulation and symplectic manifolds is followed by a description of differentiable and Riemannian manifolds to combine the classical and modern formulations. Basics of string theory, tensorial formulation of electromagnetism and general relativity are also discussed. This book adopts a practical approach and offers numerous examples and exercises for a better understanding of the subject. Required information on point set topology, calculus of variations, and multilinear algebra are provided in the appendices.


7. “MANIFOLDS AND DIFFERENTIAL GEOMETRY” by LEE J M
“Manifolds and Differential Geometry” Book Review: This book introduces the tools and structures of modern differential geometry and manifolds. Vector bundles, tensors, differential forms, de rham cohomology, the frobenius theorem and basic lie group theory are described in detail. Subsequent chapters deal with the general theory of connection on vector bundles and differential geometry of hypersurfaces in Euclidean space while also deriving the exterior calculus version of Maxwell’s equations. Fundamentals of Riemannian manifolds and Lorentz manifolds are covered under semiriemannian geometry. The book is intended for mathematicians, teachers and students studying mathematics at the graduate level.


8. “Differential Manifolds (Dover Books on Mathematics)” by Antoni A Kosinski
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“Differential Manifolds” Book Review: This book offers a comprehensive coverage of the topological structure of smooth manifolds. The book explains the technique of joining manifolds along submanifolds and the handle presentation theorem along with the proof of the hcobordism theorem based on these constructions. The Pontryagin Construction is introduced to demonstrate the link between differential topology and homotopy theory. The method of surgery is implemented for the classification of smooth structures of spheres. Basic knowledge of elementary algebraic topology is required. This book is suitable for advanced undergraduates and graduate students the systematic study of the topological structure of smooth manifolds.


9. “Differential Geometry: Manifolds, Curves, and Surfaces (Graduate Texts in Mathematics)” by Marcel Berger and Bernard Gostiaux
“Differential Geometry: Manifolds, Curves, and Surfaces” Book Review: This book covers a revised and enlarged version of the 1972 book, Geometrie Differentielle. Every topic is illustrated using nontrivial examples to familiarize the analysis and geometry in manifolds, curves, and surfaces. Separate chapters are dedicated to a detailed treatment of surfaces in threespace not from a mathematical and physical perspective.


10. “Differential Geometry: Curves – Surfaces – Manifolds” by Wolfgang Kuhnel
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“Differential Geometry: Curves – Surfaces – Manifolds” Book Review: This book provides a comprehensive introduction to differential geometry along with the recent advances made in this field. The book covers the general theory of geometry of curves and surfaces, followed by the geometry of general manifolds along with connections and curvature. Numerous figures, examples, and exercises are provided with solutions. Prerequisite knowledge of undergraduate analysis and linear algebra is required.


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