Kindly note that we have put a lot of effort into researching the best books on Differential Geometry 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" 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. “Differential Geometry of Curves and Surfaces” by M doCarmo
“Differential Geometry of Curves and Surfaces” Book Review: This book is to explain the relation between some differential formulas, like the Gauss integral for a link, or the integral of the Gaussian curvature on a surface, and topological invariants like the linking number or the Euler characteristic. This book introduces the differential geometry of curves and surfaces in both local and global aspects. This book has a different approach to the topic with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. It has many clear and well written examples along with hints to problems. It covers topics like curves, regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. This book aims to undergraduate and graduate students of mathematics and the prerequisite of the context is a brief knowledge and some familiarity with the calculus of several variables.


2. “Elementary Differential Geometry” by B O`Neill
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“Elementary Differential Geometry” Book Review: This book provides an introduction to the geometry of curves and surfaces. It emphasizes topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard along with providing a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. This book covers topics like Calculus on Euclidean Space, Frame Fields, Euclidean Geometry, Calculus on a Surface, Shape Operators, Geometry of Surfaces in R3, Riemannian Geometry, Global Structures of Surfaces. This book aims at courses, introductory courses for graduate students, individual study by mathematicians and by those in applied areas such as physics.


3. “Differential Geometry” by J J Stoker
“Differential Geometry” Book Review: This book gives a precise introduction on the topic. It makes the topic resourceful and to mathematicians as well as to the nonspecialists by the use of three different notations: vector algebra and calculus, tensor calculus.This book covers topics like Operations with Vectors, Plane Curves, Space Curves, The Basic Elements of Surface Theory, Some Special Surfaces, The Partial Differential Equations of SurfaceTheory, Inner Differential Geometry in the Small from theExtrinsic Point of View, Differential Geometry in the Large, Intrinsic Differential Geometry of Manifolds.Relativity, The Wedge Product and the Exterior Derivative ofDifferential Forms, with Applications to Surface Theory, along with two appendices for better understanding. Readers are assumed to have a passing acquaintance with linear algebra and the basic elements of analysis.
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4. “Elementary Topics in Differential Geometry” by J A Thorpe
“Elementary Topics in Differential Geometry” Book Review: This book gives a simple introduction on the topic linear algebra into the curriculum at a very sophomore level, This book has an advantage both in reading of differential equations and in the teaching of multivariate calculus. This book is good for students as completing the sophomore year now have a fair preliminary understanding of spaces of many dimensions so it is directed in such a way that students will have all the general idea and attention to the topic 2dimensional surfaces in 3space rather than to surfaces of arbitrary dimension and helps in cultivating higher dimensions about the context. This book covers topics like Graphs and Level Sets, The Tangent Space, Surfaces, The Gauss Map, Curvature of Plane Curves, Convex Surfaces, Surface Area and Volume.


5. “Differential Geometry” by Mittal
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“Differential Geometry” Book Review: This book is fully updated on the context. Its properties are analyzed in variational calculus or calculus of variations, dealing with maximization or minimization of functionality. This book has a wide range of problems and exercises that are presented and covers Definitions of Curves in space and examples, Tangent line to curve, Osculating plane, principal normal and binormal and for further reading it has many solved examples and exercises.


6. “Differential Geometry (Dover Books on Mathematics” by Erwin Kreyszig
“Differential Geometry (Dover Books on Mathematics” Book Review: This book is resourceful on the topic and has a distinguished introduction on differential geometry of curves and surfaces in threedimensional Euclidean space. The context here is presented in most simple and its most essential form with many explanatory details, figures and examples, and in such a manner that conveys the geometric significance, theoretical and practical importance of the different concepts, methods and results involved. The book focuses more on the basic concepts and facts of analytic geometry, the theory of space curves, and the foundations of the theory of surfaces, along with fundamental problems. This book addresses topics like geodesics, mappings of surfaces, special surfaces, and the absolute differential calculus and the displacement of LeviCivita. It has problems at the end of the book along with solutions for meaningfully reviewing the material presented, and familiarizes students with the manner of reasoning in differential geometry.


7. “Differential Geometry: A First Course” by D Somasundaram
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“Differential Geometry: A First Course” Book Review: This book gives a classical introduction on the topic theory of space curves and surfaces offered at the underGraduate and Postgraduate courses in Mathematics. The theory is well developed and concluded with a detailed discussion on fundamental existence theorems. The book covers topics like trinsic properties, geodesics on surfaces, the second fundamental form with local nonintrinsic properties and the fundamental equations of the surface theory with several applications. This covers the contents well even the theorems are presented well along with its proof clearly and is good for the context. This book gives a reader an illuminating insight into the essence of Mathematics and if a reader wants to reach to the comprehension of the topic then he or she can follow this book.


8. “Differential Geometry (M.Sc)” by S G Venkatachalapathy  
9. “A Course in Differential Geometry and Lie Groups (Texts and Readings in Mathematics)” by S Kumaresan
“A Course in Differential Geometry and Lie Groups (Texts and Readings in Mathematics)” Book Review: This presents the topic very well and gives a clear understanding on the topic. It covers topics like differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The book emphasizes geometric concepts, giving the reader a working knowledge of the topic. Exercises are included in this book along with Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples. There is a thorough discussion on the existence, uniqueness, and smooth dependence of solutions of ODEs. This book has a very classical treatment and a modern approach to the topic along with simple proofs to hairyball theorem, hairyball theorem. This book is suitable for a graduatelevel introduction to basic differential and Riemannian geometry.


10. “Differential Geometry of Manifolds” by Khan
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“Differential Geometry of Manifolds” Book Review: This book has well precise, natural texts on the topic which is well understood. This book is concerned with mathematical formulation and trying to answer them using calculus techniques. This book discusses the theory of differential and Riemannian manifolds to help students understand the basic structures and consequent developments. This book deals with the topic like algebra and calculus, Riemannian geometry, submanifolds and hypersurfaces, almost complex manifolds, etc., with utmost care so that maximum information can be enabled and the reader can grasp them easily. The book is for the postgraduate students of mathematics and is also useful to the researchers working in the field of differential geometry and its applications to general theory of relativity and cosmology and other applied areas. The book provides all the necessary information on the topic and has easy to understand texts with a large number of examples and illustrations at appropriate places.


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