作者: Matthias Günther
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摘要: The dot in (1) denotes the usual scalar product of R. notion embedding means, that w is locally an immersion and globally a homeomorphism M onto subspace u(M) R*. If : -• R satisfies on whole M, we speak isometric embedding. solution (possibly small) neighbourhood any point local A further question regularity dependence metric. And finally, what can be said about minimal value ? We will give some historical remarks. There exists great number beautiful papers which handle problem (local or global) under assumptions manifold metric g (e.g. special values dimension n, positivity curvature), but are interested only general problem. Janet (1926), Cartan (1927) Burstin (1931) proved existence with q = n(n + l)/2 analytical case; essentials their proofs suitable applications Cauchy-Kowalewski theorem. Up to date corresponding result (with same without assumptions) unknown nonanalytical case, even for n 2. Nash (1954) Kuiper (1955) (global) class C < 2n -f 1 (in fact results more subtle),