In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a...

We prove the existence and uniqueness of Levi-Civita connections for strongly [Formula: see text]-compatible pseudo-Riemannian metrics on tame differential calculi. Such properly contain classes bilinear as well their conformal deformations. This extends previous results in [J. Bhowmick, D. Goswami S. Mukhopadhyay, a class spectral triples, Lett. Math. Phys. 110 (2020) 835–884] G. Landi, On Kos...

In this paper, we study lightlike submanifolds of indefinite Kaehler manifolds. We introduce a class of lightlike submanifold called semi-invariant lightlike submanifold. We consider lightlike submanifold with respect to a quarter-symmetric non metric connection which is determined by the complex structure. We give some equivalent conditions for integrability of distributions with respect to th...

We study lightlike hypersurfaces of a semi-Riemannian product manifold. We introduce a class of lightlike hypersurfaces called screen semi-invariant lightlike hypersurfaces and radical antiinvariant lightlike hypersurfaces. We consider lightlike hypersurfaces with respect to a quartersymmetric nonmetric connection which is determined by the product structure. We give some equivalent conditions ...

We obtain curvature tensor R̃(X, Y )Z w.r.t quarter-symmetric metric connection in terms of curvature tensor R(X,Y )Z relative to the Levi-civita connection in a K-contact manifold. Further, locally φ-symmetric, φ-symmetric and locally projective φ-symmetric K-contact manifolds with respect to the quarter-symmetric metric connection are studied and some results are obtained. The results are assi...

The purpose of this paper is to define a diagonal lift Dg of a Riemannian metric g of a manifold Mn to the cotangent bundle T (Mn) of Mn, to associate with Dg an Levi-Civita connection of T(Mn) in a natural way and to investigate applications of the diagonal lifts.

The Dirac monopoles in 3-space and their generalization by C. N. Yang to 5-space are observed to be just the Levi-Civita spin connection of the cylindrical Riemannian metric on the 3and 5dimensional punctured spaces. Their straightforward generalization to higher dimensions is also investigated.

We prove the Focal Index Lemma and the Rauch and Berger comparison Theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop space Ω(M) with the L2 (weak) Riemannian structure.

The maximal complexifications adapted to the Levi Civita connection for a distinguished one-parameter family of left-invariant metrics on a real, non-compact, semisimple Lie group G are determined. For G = SL2(R) their realization as invariant Riemann domains over G = SL2(C) is carried out and their complex-geometric properties are investigated. One obtains new non-univalent, non-Stein examples.