We study Hermitian metrics with a Gauduchon connection being "K\"ahler-like", namely, satisfying the same symmetries for curvature as Levi-Civita and Chern connections. In particular, we investigate $6$-dimensional solvmanifolds invariant complex structures trivial canonical bundle metrics. The results this case give evidence two conjectures that are expected to hold in more generality: first, ...

A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs nets composed of foliations. In this article, we define mixed scalar curvature structure a linear connection, represent kind using fundamental tensors divergence geometrically interesting vecto...

The general relativity theory is redefined equivalently in almost Kähler variables: symplectic form, θ[g], and canonical symplectic connection, D̂[g] (distorted from the Levi–Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metri...

In the present paper a global conformal invariant Y of a closed initial data set is constructed. A spacelike hypersurface Σ in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation De, the so-called real Sen connection, which turns out to be determined completely by the initial data hab and χab induced on Σ, and coincides, in the case of vanishing second fundamen...

Given a tame differential calculus over noncommutative algebra [Formula: see text] and an text]-bilinear metric consider the conformal deformation being invertible element of We prove that there exists unique connection on bimodule one-forms which is torsionless compatible with derive concrete formula connecting Levi-Civita for As application, we compute Ricci scalar curvatures general perturba...

We carry out the programme of R. Liouville [18] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [Γ] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [Γ] or as a w...

We classify 7-dimensional cocalibrated G2-manifolds with parallel characteristic torsion and non-abelian holonomy. All these spaces admit a metric connection ∇ with totally skew-symmetric torsion and a spinor field Ψ1 solving the equations in the common sector of type II superstring theory. There exist G2-structures with parallel characteristic torsion that are not naturally reductive. 1. Metri...

In abstract Yang–Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature...