11 research outputs found

    Exotic Differentiable Structures and General Relativity

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    We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as S7S^7, \R and S3Ă—R1.S^3\times {\bf R^1}. Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe

    Real Kaehler submanifolds in low codimension

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    AbstractIsometric immersions of Kaehler manifolds into euclidean space with real codimension 1 have been completely described, in codimension 2 at least in the complete minimal case. Codimension 3 examples are immediately obtained as complex hypersurfaces of Kaehler submanifolds in codimension one. The main purpose of this paper is to show that all local examples arise this way generically

    On the almost negatively curved 33-sphere

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    Metric foliations and curvature

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