Differential Geometry of Manifolds

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Metrics of a space with linear connection which is not semi-symmetric



It is well-known Levi-Chivita’s construction of object for affine connection (in modern terminology — linear connection) by the field of non-degenerate metric on a smooth manifold. An inverse problem (a construction of metric by given linear connection) is solved ambiguously, besides, the metric may turn out to be degenerate and indefinite. On the one hand, two metrics differing in a sign are obviously build: by curvature tensor contractionwith subsequent symmetrization. Оn the other hand, Vranceanu’s metric is a double contraction of multiplication of a torsion tensor’s components. In this paper Levi-Chivita’s inverse problem is solved in other way using the field of connection object.

It is proved that in the general case, when the linear connection is not semi-symmetric, six metrics can be constructed. In the special case, when the linear connection is semi-symmetric (in particular, torsion-free), the constructed metrics vanish.

The investigation is done on a semi-holonomic smooth manifold by means of two prolongation its structure equations.


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