RUSSIAN FEDERAL COMMITTEE FOR HIGHER EDUCATION
arXiv:math.HO/0412421 v1 21 Dec 2004
BASHKIR STATE UNIVERSITY
SHARIPOV R. A.
COURSE OF DIFFERENTIAL GEOMETRY
The Textbook
Ufa 1996
2
MSC 97U20 UDC 514.7 Sharipov R. A. Course of Differential Geometry: the textbook / Publ. of Bashkir State University — Ufa, 1996. — pp. 132. — ISBN 5-7477-0129-0.
This book is a textbook for the basic course of differential geometry. It is recommended as an introductory material for this subject. In preparing Russian edition of this book I used the computer typesetting on the base of the AMS-TEX package and I used Cyrillic fonts of the Lh-family distributed by the CyrTUG association of Cyrillic TEX users. English edition of this book is also typeset by means of the AMS-TEX package. Referees:
Mathematics group of Ufa State University for Aircraft and Technology (UGATU); Prof. V. V. Sokolov, Mathematical Institute of Ural Branch of Russian Academy of Sciences (IM UrO RAN).
Contacts to author. Office: Phone: Fax:
Mathematics Department, Bashkir State University, 32 Frunze street, 450074 Ufa, Russia 7-(3472)-23-67-18 7-(3472)-23-67-74
Home: 5 Rabochaya street, 450003 Ufa, Russia Phone: 7-(917)-75-55-786 E-mails: R
[email protected] [email protected] ra
[email protected] ra
[email protected] URL: http://www.geocities.com/r-sharipov
ISBN 5-7477-0129-0 English translation
c Sharipov R.A., 1996
c Sharipov R.A., 2004
CONTENTS.
CONTENTS. ............................................................................................... 3. PREFACE. .................................................................................................. 5. CHAPTER I. CURVES IN THREE-DIMENSIONAL SPACE. ....................... 6. § 1. Curves. Methods of defining a curve. Regular and singular points of a curve. ............................................................................................ 6. § 2. The length integral and the natural parametrization of a curve. ............. 10. § 3. Frenet frame. The dynamics of Frenet frame. Curvature and torsion of a spacial curve. ............................................................................... 12. § 4. The curvature center and the curvature radius of a spacial curve. The evolute and the evolvent of a curve. ............................................... 14. § 5. Curves as trajectories of material points in mechanics. .......................... 16. CHAPTER II. ELEMENTS OF VECTORIAL AND TENSORIAL ANALYSIS. .......................................................... 18. § § § § § § § § § §
1. Vectorial and tensorial fields in the space. ............................................. 2. Tensor product and contraction. ........................................................... 3. The algebra of tensor fields. ................................................................. 4. Symmetrization and alternation. .......................................................... 5. Differentiation of tensor fields. ............................................................. 6. The metric tensor and the volume pseudotensor. ................................... 7. The properties of pseudotensors. .......................................................... 8. A note on the orientation. ..............