# Differential and Riemannian Manifolds by Serge Lang (auth.), Serge Lang (eds.) By Serge Lang (auth.), Serge Lang (eds.)

This is the 3rd model of a ebook on differential manifolds. the 1st model seemed in 1962, and was once written on the very starting of a interval of significant growth of the topic. on the time, i discovered no passable ebook for the principles of the topic, for a number of purposes. I increased the publication in 1971, and that i extend it nonetheless extra this present day. particularly, i've got additional 3 chapters on Riemannian and pseudo Riemannian geometry, that's, covariant derivatives, curvature, and a few functions as much as the Hopf-Rinow and Hadamard-Cartan theorems, in addition to a few calculus of adaptations and purposes to quantity types. i've got rewritten the sections on sprays, and i've given extra examples of using Stokes' theorem. i've got additionally given many extra references to the literature, all of this to increase the viewpoint of the ebook, which i am hoping can be utilized between issues for a common direction top into many instructions. the current publication nonetheless meets the outdated wishes, yet fulfills new ones. on the most elementary point, the e-book offers an advent to the elemental thoughts that are utilized in differential topology, differential geometry, and differential equations. In differential topology, one reports for example homotopy sessions of maps and the opportunity of discovering appropriate differentiable maps in them (immersions, embeddings, isomorphisms, etc.).

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Extra resources for Differential and Riemannian Manifolds

Example text

2 Some properties of series-parallel graphs We now show that fixed-point induction can also be used for proving universal properties of equational sets of graphs. 3: S = (S S) ∪ (S • S) ∪ {e}, where S ⊆ J2d . We will prove the assertions ∀G ∈ S. Pi (G), where the properties Pi are defined as follows: P1 (G) P2 (G) P3 (G) P4 (G) :⇐⇒ G is connected, :⇐⇒ G is bipolar, :⇐⇒ G is planar, :⇐⇒ G has no directed cycle. Adirected graph G with two sources denoted by srcG (1) and srcG (2) (cf. 3) is bipolar if it has no directed cycle and every vertex belongs to a directed path from srcG (1) to srcG (2).

The references section is organized in two parts: the first part (with reference labels starting with *) lists books, book chapters and survey articles. The second lists research articles and dissertations. All necessary definitions will be given, but the reader is expected to be familiar with the basic notions of Logic (mainly first-order logic), Universal Algebra (algebras, congruences), Formal Language Theory (context-free grammars, finite automata), and Graph Theory (basic notions). Chapters 2 to 9 present detailed proofs of results that have been published in articles.

Tρ( f ) ). This operation performs no computation; it combines its arguments which are terms into a larger term. For every F-algebra M, a term t ∈ T (F) has a value tM in M that is formally defined as follows: tM := fM if t = f and f has arity 0 (it is a constant symbol), tM := fM (t1M , . . , tρ( f )M ) if t = f (t1 , . . , tρ( f ) ). Since every term can be written in a unique way as f or f (t1 , . . , tρ( f ) ) for terms t1 , . . , tρ( f ) , the value tM of t is well defined. The mapping t → tM , also denoted by 6 For associative binary operations the more readable infix notation will be used, although it is ambiguous as already observed.