bibliography for differential geometry

bibliography for differential geometry

References for Differential Geometry, MSC 53 - XX

The following are excellent sources for the indicated areas in

differential geometry

.

Foundations

1.

Richard L. Bishop and Samuel I. Goldberg

Tensor (

http://planetmath.org/Tensor

) Analysis on Manifolds

Macmillan 1968; Dover, New York, 1980 ISBN 0-486-640439-6

Don’t be fooled by the title! Although you will find little in the way of the

index gymnastics

traditionally associated with

tensor analysis

, this book nevertheless exercise your mind in abstract thinking and lay a solid

foundation

for further study in differential geometry,

topology

, and

rational

mechanics. The prerequisites for this book are advanced

calculus

,

linear algebra

, and the basics of

differential equations

. (Also, familiarity with the abstract point of view in mathematics is a must. As the authors say in the

introduction

“An understanding of what it means for

solutions of systems

of differential equations to exist and be unique is more imporant than an ability to crank out

general solutions

.”) Starting with the basics of

set theory

, the authors develop the basics of topology and linear algebra. On this foundation, they define

manifolds

and

structures

on manifolds. They then develop the basics of the theory of exterior forms (including an introduction to such topics as integration of differential forms on simplicial chains,

de Rham cohomology

, and the formulation of systems of differential equations as Pfaffian systems), Riemannian and semi-Riemannian geometry, and culminates in a grand finale wherein the authors apply the tools of Riemannian, symplectic, and contact geometry developed in the preceding chapters to the science of mechanics. The power and utility of the theory developed in this book is amply demonstrated by the fact that in this brief chapter of 18 pages, the authors manage to prove the conservation of energy and momentum, describe the evolution of a mechanical system as the motion of a point on a manifold, and introduce such advanced topics as non-holonomic constraints and transformation theory of phase space. (Indeed, the development of classical mechanics given here is so sophisticated that one could arrive at general relativity and quantum mechanics in a few short steps!)

2.

Manfredo Perdigão do Carmo

Riemannian Geometry

,
Birkhauser, Boston, 1992 ISBN 0-8176-3490-8 ISBN 3-7643-3490-8 QA649.C2913

3.

Michael Spivak,

Calculus on Manifolds

, Mathematical Monographs Series, Addison-Wesley 1965, ISBN 0-8053-9021-9.

A basic but thorough introduction to

multivariable calculus

from the point of view of differential geometry. Suitable for very bright students, it deals only with implicitly defined manifolds but defines

differentiation

and integration in their proper generality (for example, integration is integration of differential forms over

singular

n

-chains). It proves Stokes’

theorem

in its fully general form. Requires no background beyond single-variable calculus.

Classical differential geometry, MSC 53 A

1.

Heinrich W. Guggenheimer

Differential Geometry

, McGraw-Hill, New York, 1963; Dover, New York, 1977, ISBN 0-486-63433

2.

Eisenhart, Luther Pfahler

Differential Geometry

, Princeton Unversity Press, Princton, N.J. 1925, 1977, ISBN 0-691-08026-7

Local differential geometry, MSC 53 B

1.

William M. Boothby,

An introduction to Differentiable Manifolds and Riemannian Geometry

, Second Edition, Academic Press, 1986.

This book provides the background in differential geometry required to understand general relativity from a fully

rigorous

point of view. Tensors, forms, integration and differentiation are all covered, as are

curvature

and

geodesics

. A very brief

section

at the end of the book discusses spaces of constant curvature; otherwise no global theory is included.

2.

J. A. Schouten, W. v. d. Kulk,

Pfaff’s problem and its

generalizations

Oxford University Press 1949, Chelsea Publishing, New York, 1969 ISBN 8284-0221-3

This is an comprehensive treastise which includes most everything if not everything that was known about the topic of Pfaff’s problem up to the 1940’s. The only disadvantage is that the authors employ a complicated system of index notation which includes not only

subscripts

and superscripts, but also indices above and below the main symbol and a

variety

of fonts and

alphabets

.

Global differential geometry, MSC 53 C

1.

Morgan, Frank,

Geometric Measure Theory

Academic Press, Can Diego, Ca. 1988 ISBN 0-12-506855-7 QA312.M67

Symplectic differential geometry and contact geometry, MSC 53 D

Title

bibliography for differential geometry

Canonical name

BibliographyForDifferentialGeometry

Date of creation

2013-03-22 14:16:46

Last modified on

2013-03-22 14:16:46

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

15

Author

archibal (4430)

Entry type

Bibliography

Classification

msc 53-00

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