Let (M, g) be an n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G
 and called Mus-Cheeger-Gromoll metric on tangent bundle TM. We calculate its Levi-Civita connection curvature tensor. Also, study geometry (TM, G).

In this note we discuss conditions under which a linear connection on a manifold equipped with both a symmetric (Riemannian) and a skew-symmetric (almost-symplectic or Poisson) tensor field will preserve both structures. If (M, g) is a (pseudo-)Riemannian manifold, then classical results due to T. Levi-Civita, H. Weyl and E. Cartan [7] show that for any (1, 2) tensor field T i jk which is skew-...

Recently a class of alternative theories of gravity which goes under the name f(R) gravity, has received considerable attention, mainly due to its interesting applications in cosmology. However, the phenomenology of such theories is not only relevant to cosmological scales, especially when it is treated within the framework of the so called Palatini variation, an independent variation with resp...

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...

For a given (n − 1)-dimensional hypersurface x : M → R, consider the Laguerre form Φ, the Laguerre tensor L and the Laguerre second fundamental form B of the immersion x. In this article, we address the case when the Laguerre form of x is parallel, i.e., ∇Φ ≡ 0. We prove that ∇Φ ≡ 0 is equivalent to Φ ≡ 0, provided that either L+λB+μg = 0 for some smooth function λ and μ, or x has constant Lagu...

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M . This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric constr...

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce...

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...