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A pseudo-Riemannian structure on the cotangent bundle. (English) Zbl 0758.53036
Let $M$ be an $n$-dimensional manifold and ${T}^{*}M$ its cotangent bundle. Suppose that ${T}^{*}M$ is endowed with a symmetric nonlinear connection of local coefficients ${N}_{ij}$. This paper studies some properties of the pseudo-Riemannian metric $G=2\left(d{p}_{i}+{N}_{ij}d{q}^{j}\right)d{q}^{i}$ on ${T}^{*}M$, where $q=\left({q}^{i}\right)$, $p=\left({p}_{i}\right)$, $\left(q,p\right)\in {T}^{*}M$. The most important theorems refer to Levi-Civita connection, parallel transport and curvature tensor field.

##### MSC:
 53C50 Lorentz manifolds, manifolds with indefinite metrics