Abstract
This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi–Civita connection.
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Acknowledgements
The author is indebted to Lars Hesselholt and Yuri I. Manin for inspiring suggestions. The presentworkwas partially supported by the Max-Planck-Institut für Mathematik in Bonn, by the Institut des Hautes Études Scientifiques in Bures sur Yvette, and by the Simons Foundation (Award 311773).
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