On some differential operators on natural Riemann extensions

Abstract

Natural Riemann extensions are pseudo-Riemannian metrics (introduced by Sekizawa and studied then by Kowalski-Sekizawa), which generalize the classical Riemann extension defined by Patterson-Walker. Let M be a manifold with an affine connection and let T * M be the total space of its cotangent bundle. On T * M endowed with a natural Riemann extension, we study here the Laplacian and give necessary and sufficient conditions for the harmonicity of a certain family of (local) functions. We also prove a gradient formula for natural Riemann extensions.

Almost Para-Hermitian and Almost Paracontact Metric Structures Induced by Natural Riemann Extensions

“…For later use, we recall that the gradient of a real function F : N −→ R on a (semi-) Riemannian manifold (N, h) is given by h(gradF, X) = dF (X), X ∈ χ(N ) and h is a (semi-) Riemannian metric on N . In [4] the following formula for the gradient of the vertical lift Z V on T * M of Z ∈ χ(M ) with respect to the proper natural Riemann extension g on T * M is obtained:…”

“…In Sect. 4 we study a family of non-degenerate hypersurfaces H t of (T * M, P, g). They are a generalization of the family H t of non-degenerate hypersurfaces of (T * M, g), introduced in [3].…”

A Harmonic Endomorphism in a Semi-Riemannian Context

“…We construct as in [5], an orthonormal basis {E i , E i * } i=1,...,n with respect toḡ in T (x,w) (T * M ) which is defined at any point (x, w) ∈ T * M by the formulas…”

“…In [12], Patterson and Walker introduced the (classical) Riemann extension that was generalized by Sekizawa and Kowalski to natural Riemann extension, which is a semi-Riemannian metric of signature (n, n), on the total space of T * M , (see [14] and [11]). Later, Bejan and Kowalski [5] characterized harmonic functions with respect to the natural Riemann extensionḡ on T * M . Also, the natural Riemann extension is a special class of modified Riemann extensions which is studied in [7] and [10].…”

A characterization of the Riemann extension\newline in terms of harmonicity

“…Natural Riemann extension is a special class of both the modified Riemann extension (see [6] and [9]) and the general Riemann extension. Bejan and Kowalski characterized in [5] some harmonic functions on (T * M, g).…”