---
product_id: 7124353
title: "Riemannian Geometry"
price: "E£ 7158"
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# Riemannian Geometry

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- **What is this?** Riemannian Geometry
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Review: Excellent stepping stone to more advanced treatments - 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 more 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?
Review: good, instructive book, but not for the "beginners" - 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 more pictures but it's not to the level that bothers me. Overall, great book to read on your own!

## Technical Specifications

| Specification | Value |
|---------------|-------|
| Best Sellers Rank | #391,944 in Books ( See Top 100 in Books ) #19 in Analytic Geometry (Books) #24 in Differential Geometry (Books) #127 in Mathematical Physics (Books) |
| Customer Reviews | 4.7 4.7 out of 5 stars (93) |
| Dimensions  | 7.4 x 0.7 x 8.7 inches |
| Edition  | First Edition |
| ISBN-10  | 0817634908 |
| ISBN-13  | 978-0817634902 |
| Item Weight  | 2.31 pounds |
| Language  | English |
| Print length  | 300 pages |
| Publication date  | January 1, 1992 |
| Publisher  | Springer |

## Images

![Riemannian Geometry - Image 1](https://m.media-amazon.com/images/I/51pbdXHKO-L.jpg)

## Customer Reviews

### ⭐⭐⭐⭐⭐ Excellent stepping stone to more advanced treatments
*by P***K on June 13, 2016*

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 more 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?

### ⭐⭐⭐⭐⭐ good, instructive book, but not for the "beginners"
*by J***Y on May 31, 2013*

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 more pictures but it's not to the level that bothers me. Overall, great book to read on your own!

### ⭐⭐⭐⭐⭐ Very good and classic book
*by S***I on June 26, 2024*

If you have studied the author's another famous textbook differential geometry of curves and understand it well, you can start to read this book. Do Carmo go straight to the centre of Riemannian Geometry, so you should get the essence of this subject if you study it in detail and patiently.

## Frequently Bought Together

- Riemannian Geometry
- Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition (Dover Books on Mathematics)
- Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218)

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