Riemannian Geometry Is An Expanded Edition Of A Highly Acclaimed And Successful Textbook Originally Published In Portuguese For First Year Graduate Students In Mathematics And Physics The Author S Treatment Goes Very Directly To The Basic Language Of Riemannian Geometry And Immediately Presents Some Of Its Most Fundamental Theorems It Is Elementary, Assuming Only A Modest Background From Readers, Making It Suitable For A Wide Variety Of Students And Course Structures Its Selection Of Topics Has Been Deemed Superb By Teachers Who Have Used The Text A Significant Feature Of The Book Is Its Powerful And Revealing Structure, Beginning Simply With The Definition Of A Differentiable Manifold And Ending With One Of The Most Important Results In Riemannian Geometry, A Proof Of The Sphere Theorem The Text Abounds With Basic Definitions And Theorems, Examples, Applications, And Numerous Exercises To Test The Student S Understanding And Extend Knowledge And Insight Into The Subject Instructors And Students Alike Will Find The Work To Be A Significant Contribution To This Highly Applicable And Stimulating Subject

7 thoughts on “Riemannian Geometry: Theory & Applications (Mathematics: Theory & Applications)”

Este libro es la traducci n en los primeros setenta de un texto en portugu s Su principal inter s reside en el hincapi que hace en destacar el papel que juegan las distintas hip tesis y condiciones en los teoremas de la geometria diferencial Si alguien quiere entender lo que est haciendo al manipular s mbolos este es el libro.

Though this text lacks a categorical flavor with commutative diagrams, pull backs, etc it is still at an intermediate to advanced level Nevertheless, constructs are developed which are assumed in a categorical treatment It does do Hopf Rinow, Rauch Comparison, and the Morse Index Theorems which you would find in a text like Bishop Crittendon However, it does the Sphere Theorem, an advanced theorem dependent on the Morse Theory calculus of variations methods in differential geometry Even energy is treated which is the kinetic energy functional integral used to determine minimal geodesics, reminiscent of the Maupertuis Principle in mechanics.The reader is assumed to be familiar with differentiable manifolds but a somewhat scant Chapter 0 is given which mostly collects results which will be needed later The treatment is dominated by the coordinate free approach so emphasis is on the tangent plane or space and properties intrinsic to the surface with only a brief section on tensor methods given Realize the tangent space has the same dimension as the surface to which it is tangent and this can be greater than 2 If you remember from advanced calculus, you took the gradient of a function of n variables the function maps to a constant as a sphere say does The gradient defined the normal to the n 1 dimensional tangent hyperplane to the surface The surface is also n 1 dimensional since given n 1 values to the variables the nth value is determined by the function equation implicitly Note in this construction we used the embedding in our interpretation, nevertheless this gradient tangent hyperplane notion can be given an intrinsically defined method of getting the tangent space through the related notion of the directional derivative Forging this to a linear tangent space is a key construct which the reader should grasp, one not available in Gauss s lifetime The text by Boothby is user friendly here and is also available online as a free PDF Boothby essentially covers the first five chapters of do Carmo including Chapter 0 filling in many of the gaps.Both in Boothby and do Carmo the affine connection makes appearance axiomatically and the covariant derivative results from imposed conditions in a theorem construct If this is a bit hard to chew it was for me there are exercises 1 and 2 on pp 56 57 of do Carmo in which you are to show how the affine connection and covariant derivative arise from parallel transport Theorem 3.12 of Chapter VII in Boothby does this a bit too formally but you can find it in various forms on the web In particular there is a nice one where the tangent planes are related along the curve over which the parallel transport or propagation occurs resulting in a differential equation which gives both the affine connection and the covariant derivative Just Google parallel transport and covariant derivative.I have certain quibbles like in defining the Riemannian metric as a bilinear symmetric form,i.e., his notation is a bit dated here and there but the text shines from chapter 5 on So 5 stars.P.S There s a PDF entitled An Introduction to Riemannian Geometry by Sigmundur Gudmundsson which is free and short and is tailor made for do Carmo assuming only advanced calculus as in say rigorous proof of inverse function theorem or the first nine or ten chapters of Rudin s Principles 3rd It does assume some familiarity with differential geometry in R 3 as in do Carmo s earlier text but you can probably fill this in from the web if you re not familiar from past coursework as in vector analysis Differential manifold and tangent space are clearly developed without the topological detours pretty much if you re familiar with the derivative as a linear map as in Rudin , you re at the right level Also Lang s Introduction to Differentiable Manifolds is available as a free PDF if you want to see the categorical treatment after you get through do Carmo can also be used for reference concurrently, example isomorphic linear spaces

This is the book isn t for someone who has never been exposed to differential geometry If you know the basics of manifolds and are determined to learn some fairly difficult mathematics this is the book to learn Riemannian Geometry from.

There are quite a few pages where the font changes abruptly from Bookman Old Style to Arial font, very bizarre Otherwise, pretty good the binding seems solid.

This is a concise, and instructive book that can be read easily.However, this is not for the absolute beginner Let me explain what kinds of knowledge you should have before digging into this book You should already be familiar with basic smooth manifold theory found in first few chapters of books such as Introduction to Smooth Manifolds by John Lee For example, the author assumes that you already know how to define tangent space using derivation He also assumes that you know the precisely how to show maps between two manifolds are smooth using the coordinate presentation of the map He also assumes you know tensors He won t really distinguish coordinate presentation vs the actual map because these are all assumed to be already mastered by the reader.Also, you have to be able to understand his notation from the proof He has his own set of notations without explanation Once you read the proof, its meaning becomes clear but this won t happen unless you have some knowledge in smooth manifold theory.With all these prerequisite, reading should be smooth and fun I sometimes wished he had pictures but it s not to the level that bothers me Overall, great book to read on your own

Este libro es la traducci n en los primeros setenta de un texto en portugu s Su principal inter s reside en el hincapi que hace en destacar el papel que juegan las distintas hip tesis y condiciones en los teoremas de la geometria diferencial Si alguien quiere entender lo que est haciendo al manipular s mbolos este es el libro.

Todo perfecto

Though this text lacks a categorical flavor with commutative diagrams, pull backs, etc it is still at an intermediate to advanced level Nevertheless, constructs are developed which are assumed in a categorical treatment It does do Hopf Rinow, Rauch Comparison, and the Morse Index Theorems which you would find in a text like Bishop Crittendon However, it does the Sphere Theorem, an advanced theorem dependent on the Morse Theory calculus of variations methods in differential geometry Even energy is treated which is the kinetic energy functional integral used to determine minimal geodesics, reminiscent of the Maupertuis Principle in mechanics.The reader is assumed to be familiar with differentiable manifolds but a somewhat scant Chapter 0 is given which mostly collects results which will be needed later The treatment is dominated by the coordinate free approach so emphasis is on the tangent plane or space and properties intrinsic to the surface with only a brief section on tensor methods given Realize the tangent space has the same dimension as the surface to which it is tangent and this can be greater than 2 If you remember from advanced calculus, you took the gradient of a function of n variables the function maps to a constant as a sphere say does The gradient defined the normal to the n 1 dimensional tangent hyperplane to the surface The surface is also n 1 dimensional since given n 1 values to the variables the nth value is determined by the function equation implicitly Note in this construction we used the embedding in our interpretation, nevertheless this gradient tangent hyperplane notion can be given an intrinsically defined method of getting the tangent space through the related notion of the directional derivative Forging this to a linear tangent space is a key construct which the reader should grasp, one not available in Gauss s lifetime The text by Boothby is user friendly here and is also available online as a free PDF Boothby essentially covers the first five chapters of do Carmo including Chapter 0 filling in many of the gaps.Both in Boothby and do Carmo the affine connection makes appearance axiomatically and the covariant derivative results from imposed conditions in a theorem construct If this is a bit hard to chew it was for me there are exercises 1 and 2 on pp 56 57 of do Carmo in which you are to show how the affine connection and covariant derivative arise from parallel transport Theorem 3.12 of Chapter VII in Boothby does this a bit too formally but you can find it in various forms on the web In particular there is a nice one where the tangent planes are related along the curve over which the parallel transport or propagation occurs resulting in a differential equation which gives both the affine connection and the covariant derivative Just Google parallel transport and covariant derivative.I have certain quibbles like in defining the Riemannian metric as a bilinear symmetric form,i.e., his notation is a bit dated here and there but the text shines from chapter 5 on So 5 stars.P.S There s a PDF entitled An Introduction to Riemannian Geometry by Sigmundur Gudmundsson which is free and short and is tailor made for do Carmo assuming only advanced calculus as in say rigorous proof of inverse function theorem or the first nine or ten chapters of Rudin s Principles 3rd It does assume some familiarity with differential geometry in R 3 as in do Carmo s earlier text but you can probably fill this in from the web if you re not familiar from past coursework as in vector analysis Differential manifold and tangent space are clearly developed without the topological detours pretty much if you re familiar with the derivative as a linear map as in Rudin , you re at the right level Also Lang s Introduction to Differentiable Manifolds is available as a free PDF if you want to see the categorical treatment after you get through do Carmo can also be used for reference concurrently, example isomorphic linear spaces

This is the book isn t for someone who has never been exposed to differential geometry If you know the basics of manifolds and are determined to learn some fairly difficult mathematics this is the book to learn Riemannian Geometry from.

There are quite a few pages where the font changes abruptly from Bookman Old Style to Arial font, very bizarre Otherwise, pretty good the binding seems solid.

This book is worth reading

This is a concise, and instructive book that can be read easily.However, this is not for the absolute beginner Let me explain what kinds of knowledge you should have before digging into this book You should already be familiar with basic smooth manifold theory found in first few chapters of books such as Introduction to Smooth Manifolds by John Lee For example, the author assumes that you already know how to define tangent space using derivation He also assumes that you know the precisely how to show maps between two manifolds are smooth using the coordinate presentation of the map He also assumes you know tensors He won t really distinguish coordinate presentation vs the actual map because these are all assumed to be already mastered by the reader.Also, you have to be able to understand his notation from the proof He has his own set of notations without explanation Once you read the proof, its meaning becomes clear but this won t happen unless you have some knowledge in smooth manifold theory.With all these prerequisite, reading should be smooth and fun I sometimes wished he had pictures but it s not to the level that bothers me Overall, great book to read on your own