島根大學

ISSN:0488-6542

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島根大学論集. 自然科学 16

1966-12-25 発行

System of Local Loops on a Manifold and Affine Connection

Kikkawa, Michihiko

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The concepts of topological loops (Hofmann [2]) and analytic loops (Malcev [5]) lead us to the concept of differentiable local loops (§ 2, Definition 1) and local loops on manifolds have been studied by the author ([3]). Namely, in a differentiable manifold with an affine connection, each point has a neighbourhood which is a differentiable local loop with a binary operation defined by means of the parallel displacement of geodesics ([3] Theorem 1).

In the present paper, differentiable manifold with a system which assigns to each point a neighbourhood with a structure of local loop will be introduced (§ 2, Definition 2) and it will be shown that an affine connection of a manifold is determined by such a system (§ 3, Theorem 1). In particular, it will be proved that if a differentiable manifold M with an affine connection г is given then г coincides with the affine connection г_∑ of M which is determined by the system ∑ of local loops associated with г (Theorem 2).

In the present paper, differentiable manifold with a system which assigns to each point a neighbourhood with a structure of local loop will be introduced (§ 2, Definition 2) and it will be shown that an affine connection of a manifold is determined by such a system (§ 3, Theorem 1). In particular, it will be proved that if a differentiable manifold M with an affine connection г is given then г coincides with the affine connection г_∑ of M which is determined by the system ∑ of local loops associated with г (Theorem 2).

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AN0010814X

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