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Almost symmetric and antisymmetric spaces with an affine connection. (English. Russian original) Zbl 0938.53023
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 114-116 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 4, 454-455 (1994).
A space $$(M,\nabla)$$ with an affine connection $$\nabla$$ is called almost symmetric if the composition of two geometric symmetries is a local automorphism of the connection $$\nabla$$. Suppose that $$(M, \nabla)$$ is almost symmetric. Denote by $$R$$ and $$T$$ its curvature tensor and torsion tensor, respectively. These tensors define the ternary operation $$\xi,\eta\zeta \rightarrow (\xi,\eta\zeta) = R(\xi,\eta)\zeta$$ and a binary operation $$\xi,\eta \rightarrow \xi \cdot \eta = \frac{1}{2} T(\xi,\eta)$$ in the tangent space $$T_e (M), e \in M$$. These two operations are connected by the following identity $$(\xi,\eta,\zeta \cdot \kappa) = (\xi,\eta,\zeta) \cdot \kappa + \zeta \cdot (\xi,\eta,\kappa)$$. Thus, $$(M,\nabla)$$ is a Lie triple system in which the binary operation $$\xi \cdot \eta$$ is defined. The authors prove that in a neighborhood of an arbitrary point $$e\in M$$, an almost symmetric space $$(M,\nabla)$$ is uniquely defined by these ternary and binary operation connected by the above mentioned identity. They also prove a similar result for an antisymmetric space with an affine connection.
##### MSC:
 53C35 Differential geometry of symmetric spaces 53C05 Connections, general theory 22A30 Other topological algebraic systems and their representations 20N05 Loops, quasigroups 17D99 Other nonassociative rings and algebras 22A22 Topological groupoids (including differentiable and Lie groupoids)