Connections which are harmonic with respect to general natural metrics

The projective curvature of the tangent bundle with natural diagonal metric

“…Replacing the above value of c 1 into the expression of P ∂ ∂y i , ∂ ∂y j ∂ ∂y k , ∀i, j, k = 1, n, this becomes of the form and then the value of d 2 is the one given by the relation (13) if and only if 3(c 2 + c 2 t) 3 (c 3 1 − 6cc 2 1 c 2 t + 14c 2 c 1 c 2 2 t 2 − 12c 3 c 3 2 t 3 ) c 2 2 t 2 (−c 1 + 4cc 2 t) 3…”

Sasaki metric on the tangent bundle of a Weyl manifold

“…Sasaki introduced in [14] his well-known Riemannian metric on T M to study some geometric properties of T M endowed with the Sasaki metric. Some extensions of the Sasaki metric were constructed on T M by Abbassi and Sarih [1,2], Janyska [8], Kowalski and Sekizawa [10], Oproiu and Papaghiuc [13], Munteanu [11], Bejan and Druta-Romaniuc [4].…”

Harmonic metrics on four dimensional non-reductive homogeneous manifolds

“…This topic was initiated by harmonic functions, i.e., real C 2 functions belonging to the kernel of the Laplace operator, and then generalized to harmonic maps between (pseudo-)Riemannian manifolds, i.e., maps which satisfy the Euler-Lagrange systems, and harmonic exterior forms, i.e., forms that are simultaneously closed and co-closed. Today, harmonicity has been extended in several directions such as harmonic morphisms (see [1]), harmonic (pseudo-)Riemannian metrics (see [11]), harmonic sections (see [5], [12]), harmonic endomorphisms (see [2], [3]), harmonic connections (see [2]) etc.…”