A note on variational and metrizable connections

Abstract: We treat the problem of metrizability of a linear connection in the context of the inverse problem of the calculus of variations. Particularly, we show that in a 2-dimensional manifold endowed with a nowhere flat connection, Helmholtz conditions can be used to answer the metrizability problem in terms of components of the connection. In dim M>=3, the question is open.

Keywords: Manifold, connection, metric, calculus of variations, inverse problem, Helmholtz conditions

Classification (MSC2000): 53C05; 53C60

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