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Applications of local algebras of differentiable manifolds. (English. Russian original) Zbl 1317.13057

J. Math. Sci., New York 207, No. 3, 485-511 (2015); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 126, 219-261 (2013).
Summary: This paper is devoted to some applications of local algebras in geometry. We recall some properties of free finite-dimensional modules over local algebras of a certain type, so-called \(A\)-spaces in the sense of Macdonald, where \(A\) denotes the algebra considered. Using these properties, we study bilinear, special symmetric, and symplectic forms on \(A\)-spaces and obtain some their invariants. Properties of these spaces are used in the study of projective Klingenberg spaces over the ring \(A\). We present fundamental notions of points and subspaces of projective Klingenberg spaces. We examine the neighborship property of points and homologies. We obtain a criterion of projective equivalence of quadrics on these spaces.

MSC:

13H05 Regular local rings
58A99 General theory of differentiable manifolds
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