Let G be a subgroup of SO(n). A G-structure on a smooth manifold M of dimension n induces a Riemannian metric g on M . The failure of the holonomy group of the Levi-Civita connection of g to reduce to G is measured by the so-called intrinsic torsion τ . It is known that the latter is a tensor which takes values at each point in T ∗ ⊗ g where T ∗ is the cotangent space and g is the orthogonal co...

After recalling some background, we define Riemannian metrics and Riemannian manifolds. We analyze the basic tensorial operations that become available in the presence of a Riemannian metric. Then we construct the Levi-Civita connection, which is the basic " new " differential operator coming from such a metric. Background The purpose of this section is two–fold. On the one hand, we want to rel...

Here the matrix (gij) (assumed nondegenerate) defines a pseudo-Riemannian metric (with upper indices) of zero curvature on the u-space, Fjk i = Fjki(u) being the corresponding Levi-Civita connection. Thus, the integrability condition can be formulated in terms of the differential geometry of SHT. For such integrable systems S. P. Tsarev [3] found a generalization (for N _> 3) of the hodograph m...

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of (J2 = ±1)-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among con...

In this paper the Berry and Aharonov-Bohm phases are generalized to nonlinear topological phase fields on pseudospheres, where the coordinate vector field is parallel transported along the signal/soliton vector field with Levi–Civita connection. Projective PSL(2,R) symmetry describes the relativistic self-interacting bosonic sine-Gordon field. A Coulomb potential can be induced as the stereogra...

Abstract. This paper presents a mathematical view of aspects of physics, showing how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. This work is motivated by discrete calculus, as it is shown that classical discrete calculus embeds in a non-commutative context. It is shown how various processe...

We investigate bi–Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non–stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric–induced nonlinear connection (N–connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a ...

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...

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with ma...

∇XV = λX for every vector field X. (1.2) Here ∇ denotes the Levi-Civita connection of g. We call vector fields satisfying (1.2) closed conformal vector fields. They appear in the work of Fialkow [3] about conformal geodesics, in the works of Yano [7–11] about concircular geometry in Riemannian manifolds, and in the works of Tashiro [6], Kerbrat [4], Kühnel and Rademacher [5], and many other aut...