We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying the existence unique Levi-Civita connection such calculus, we prove version Bianchi identity. A Gauss-Bonnet theorem for canonical calculus rank two is studied.

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFTtype correspondence between classical 2 + 1 dimensional vacuum general relativity theory on Σ× R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ, a compact, oriented Riemann surface of genus greater than one. More precisely, we prove that if over Σ with a fi...

The holonomy group of a metric g at a point p of a manifold M is the group of all linear transformations in the tangent space of p defined by parallel translation along all possible loops starting at p 1 . It is obvious that a connection can only be the Levi-Civita connection of a metric g if the holonomy group is a subgroup of the generalized orthogonal group corresponding to the signature of ...

This paper considers 4-dimensional manifolds upon which there is a Lorentz metric h and a symmetric connection Γ and which are originally assumed unrelated. It then derives sufficient conditions on h and Γ (expressed through the curvature tensor of Γ) for Γ to be the Levi-Civita connection of some (local) Lorentz metric g and calculates the relationship between g and h. Some examples are provid...

Abstract A new derivation of the flow metrics in Type IIA is given. It better adapted to formulation as a variant Laplacian flow, and it uses projected Levi–Civita connection themselves instead their conformal rescalings.

The bounded orbital motion of a massive spinless test particle in the background of a Kerr Brans-Dicke geometry is analysed in terms of worldlines that are auto-parallels of different metric compatible spacetime connections. In one case the connection is that of Levi-Civita with zero-torsion. In the second case the connection has torsion determined by the gradient of the Brans-Dicke background ...

Let M be a 2n dimensional smooth closed oriented manifold. Let g be a Riemmian metric on TM and ∇ the associated Levi-Civita connection. Let V be a complex vector bundle over M with a Hermitian metric h and a unitary connection ∇ . Let ΛC(T ∗M) be the complexified exterior algebra bundle of TM and let 〈 , 〉ΛC(T∗M) be the Hermitian metric on ΛC(T ∗M) induced by g . Let dv be the Riemannian volum...

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

A natural connection with torsion is defined, and it called the first on Riemannian Π-manifold. Relations between introduced Levi–Civita are obtained. Additionally, relations their respective curvature tensors, Ricci scalar curvatures in main classes of a classification Π-manifolds presented. An explicit example dimension five provided.