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**An introduction to differential geometry with applications to elasticity.
Reprinted from the Journal of Elasticity 78–79, No. 1-3 (2005).**
*(English)*
Zbl 1100.53004

Dordrecht: Springer (ISBN 1-4020-4247-7/hbk). 209 p. (2005).

As announced in the title of the book under review this is a book about differential geometry and elasticity theory also published earlier as journal article. And, indeed it covers both subjects in a coextensive way that can not be found in any other book in the field. Formally it is structured in four chapters – two devoted to differential geometry and the two to elasticity theory.

The first chapter deals with three-dimensional differential geometry and culminates with a proof of the continuity of an immersion as a function of the metric tensor. The next one is concerned with the differential geometry of the surfaces in the ordinary Euclidean space. The third chapter discusses direct applications of the geometry exposed in the first chapter. Finally, the fourth chapter explains the theory of shells on the basis of the results in classical differential geometry regarding surface theory exposed in the second one.

As one is immediately convinced, the above logical structure is perfect and the only drawback is the absence of any concrete examples. As a compensation the list of references containing more than 120 items is representative enough and the interested reader should be able to find them among these.

The first chapter deals with three-dimensional differential geometry and culminates with a proof of the continuity of an immersion as a function of the metric tensor. The next one is concerned with the differential geometry of the surfaces in the ordinary Euclidean space. The third chapter discusses direct applications of the geometry exposed in the first chapter. Finally, the fourth chapter explains the theory of shells on the basis of the results in classical differential geometry regarding surface theory exposed in the second one.

As one is immediately convinced, the above logical structure is perfect and the only drawback is the absence of any concrete examples. As a compensation the list of references containing more than 120 items is representative enough and the interested reader should be able to find them among these.

Reviewer: Ivailo Mladenov (Sofia)

### MSC:

53A05 | Surfaces in Euclidean and related spaces |

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

74B05 | Classical linear elasticity |

74B20 | Nonlinear elasticity |