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Embeddings (Mathematics) --- Manifolds (Mathematics)

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Concordances (Topology) --- Embeddings (Mathematics) --- Piecewise linear topology

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Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, ngeqslant 2, and let L^k be the k-th tensor power of a CR complex line bundle L over X. Given qin {0,1,ldots ,n-1}, let Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For lambda geq 0, let Pi ^{(q)}_{k,leq lambda} :=E((-infty ,lambda ]), where E denotes the spectral measure of Box ^{(q)}_{b,k}. In this work, the author proves that Pi ^{(q)}_{k,leq k^{-N_0}}F^*_k, F_kPi ^{(q)}_{k,leq k^{-N_0}}F^*_k, N_0geq 1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of Box ^{(q)}_{b,k}, where F_k is some kind of microlocal cut-off function. Moreover, we show that F_kPi ^{(q)}_{k,leq 0}F^*_k admits a full asymptotic expansion with respect to k if Box ^{(q)}_{b,k} has small spectral gap property with respect to F_k and Pi^{(q)}_{k,leq 0} is k-negligible away the diagonal with respect to F_k. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S^1 action.

Embedding theorems. --- CR submanifolds. --- Manifolds (Mathematics) --- Embeddings (Mathematics) --- Kernel functions. --- Asymptotic expansions.

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Topological embeddings

Algebraic topology --- Topological imbeddings. --- Topological imbeddings --- Embeddings, Topological --- Imbeddings, Topological --- Topological embeddings --- Embeddings (Mathematics) --- Variétés topologiques

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Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time-1-maps of Hamiltonian flows. The spectacular rigidity phenomena for symplectic mappings discovered in the last two decades show that certain things cannot be done by a symplectic mapping. For instance, Gromov's famous ""non-squeezing'' theorem states that one cannot map a ball into a thinner cylinder by a symplectic embedding. The aim of this book is to show that certain other things can be done by symplectic mappings. This is achieved by various elementary and explicit symplectic embedding.

Symplectic geometry. --- Embeddings (Mathematics) --- Géométrie symplectique --- Plongements (Mathématiques) --- Imbeddings (Mathematics) --- Geometry, Algebraic --- Immersions (Mathematics) --- Geometry, Differential