By Wilfred Kaplan
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Additional resources for Introduction to analytic functions
9) Proof. 10). 11) where fi(u) is an arbitrary smooth covector field on the manifold M. 12) where f(u) is an arbitrary function on the manifold. 11) to be closed. 15) means that the 1-form fi(u)dui on the manifold is exact. 16) where fik(u) is an arbitrary smooth covariant two-valent tensor field on the manifold M. 16) to be closed. 19) just mean that fijdui /\ duj is a closed 2-form on the manifold M. From formula (3. 20) that is, provided that fijdui Λduj is an exact 2-form on the manifold M.
40) in the case of a non-degenerate metric gij(u) and therefore it is automatically fulfilled for the metric of constant Riemannian curvature (in the case of a degenerate metric, this relation is non-trivial). 36) defines a one-component Poisson structure for any K. 7, assuming conditionally that it is possible to attribute any value of curvature K to any one-component metric. 101)) differ from each other namely by the sign of the “conditional curvature” K. 4). 36) can be rewritten in the form where is the covariant derivative generated by a metric of constant Riemannian curvature K.
The converse is also obvious: if all the tensors are equal to zero, then all the connections are equal to each other, and consequently all of them are equal to zero in the flat coordinates of the metric gij1, and all the metrics are constant in these coordinates by virtue of their compatibility with the corresponding connections. 3). 4) to be a non-degenerate Poisson structure were found. 4) that guarantee skew-symmetry of the bracket and the fulfilment of the Jacobi identity for arbitrary functionals.
Introduction to analytic functions by Wilfred Kaplan