By Serge Lang (auth.)

The current quantity supersedes my advent to Differentiable Manifolds written many years again. i've got increased the publication significantly, together with such things as the Lie spinoff, and particularly the fundamental integration thought of differential varieties, with Stokes' theorem and its a number of particular formulations in numerous contexts. The foreword which I wrote within the prior publication continues to be fairly legitimate and wishes purely moderate extension the following. among complex calculus and the 3 nice differential theories (differential topology, differential geometry, usual differential equations), there lies a no-man's-land for which there exists no systematic exposition within the literature. it's the function of this e-book to fill the distance. the 3 differential theories are not at all autonomous of one another, yet continue in line with their very own style. In differential topology, one reports for example homotopy sessions of maps and the potential for discovering compatible differentiable maps in them (immersions, embeddings, isomorphisms, etc.). One can also use differentiable constructions on topological manifolds to figure out the topological constitution of the manifold (e.g. it los angeles Smale [26]).

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**Example text**

Then the map Tf is given by Tf(x, v) = (f(x), j'(x)v) for x E X and vEE. Since j' is of class OP-l by definition, we can apply Proposition 16 loco cit. to conclude that Tf is also of class CP-l. The functoriality property is trivially satisfied, and we have therefore defined the functor T as promised. It will sometimes be notationally convenient to write f* instead of Tf for the induced map, which is also called the tangent map. The bundle T(X) is called the tangent bundle of X. §3. Exact sequences of bundles Let X be a manifold.

Then: (i) f is an immersion at x if and only if the map T xf is injective and splits. (ii) f is a submersion at x if and only if the map T xf is surjective and its kernel splits. Note. If X, Yare finite dimensional, then the condition that Txf splits is superfluous. Every subspace of a finite dimensional vector space splits. Example. Let E be a (real) Hilbert space, and let (x, y) E R be its inner product. Then the square of the norm f(x) = (x, x) is obviously of class Coo. The derivative j'(x) is given by the formula j'(x)y = 2(x, y) 28 [11,2] MANIFOLDS and for any given x #: 0, it follows that the derivative I'(x) is surjective.

Proposition 1. Let E, F be finite dimensional veckA' spaces. Let U be open in some Banach space. Let f: U x E -. F be a morphism such that for each x E U, the mo,p given by fz(v) = f(x, v) is a linear mo,p. Then the mo,p of U into L(E, F) given by x 1-+ fz is a morphism. [III, 1] DEFINITION, PULL-BACKS 43 Proof. We can write F = Rl X ••• x Rn (n copies of R). Using the fact that L(E, F) = L(E, Rl) x ... x L(E, R n), it will suffice to prove our assertion when F = R. Similarly, we can assume that E = R also.