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...

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...

We develop tools for studying the control of underactuated mechanical systems that evolve on a configuration space with a principal fiber bundle structure. Taking the viewpoint of affine connection control systems, we derive reduced formulations of the Levi-Civita and the nonholonomic affine connections, along with the symmetric product, in the presence of symmetries and nonholonomic constraint...

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence. Equivalent represent...

We present a consistent and complete description of the coupling to matter in Teleparallel Equivalent General Relativity (TEGR) theory built from Cartan connection, as we proposed previous works. A first theorem allows us obtain parallel transport connection into proper Ehresmann while second ensures link TEGR-Cartan one-form that contains Levi-Civita connection. This yields agreement with obse...

This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton’s equations, structure of Levi–Civita connection, and generalizations electromagnetism that related to theory with early work Hermann Weyl. The territory here explored is self-contained mathematically. It elementary, algebraic, su...

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with deformed Sasaki metric. In this paper, firstly we investigate all forms of curvature tensors (Riemannian tensor, Ricci curvature, sectional scalar curvature). Secondly, study the geometry unit metric, where presented formulas Levi-Civita connection also