Embeddings and immersions in Riemannian geometry

This article is a significantly expanded version of a paper read by one of the authors to the Moscow Mathematical Society [18]. It consists of two chapters and eleven appendices. The first chapter contains a survey of known results, as a rule with precise statements and references, but without full proofs. In the second chapter, the fundamental embedding theorems are set out in detail, among them some new results concerning a bound for the smallest dimension of a euclidean space in which any compact riemannian manifold of given dimension can be embedded, and also to the corresponding local problem. In Appendices 1, 3, 4, 5, 7, 8 and 9, proofs of miscellaneous propositions in the survey are given. Appendices 2 and 6 are needed for other appendices, but are interesting in their own right. In Appendix 10 a more general embedding problem is considered. Appendix 11 has a bearing on Chapter 2 and is of an analytic nature.
We wish to thank I. A. Bakel'man, A. L. Verner, Yu. A. Volkov, S. P. Geisberg, V. L. Eidlin and Ya. M. Eliashberg for their help in the preparation of the article. We are especially grateful to Yu. D. Burago who at our request modified one of his inequalities to suit our requirements (Appendix 2).