We give a method based on an idea of O. Veblen which gives explicit formulas for the covariant derivatives of natural objects in terms of the Christoffel symbols of a symmetric Ehresmann -connection.
Consider a (1,1) tensor field J, defined on a real or complex m-dimensional manifold M, whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions , defined around p, such that and , j = 1,...,m. Then there exists a dense open set such that we can find coordinates, around each of its points, on which J is written with affine coefficients. This result is obtained by associating to J a bihamiltonian structure on T*M.
Y. Euh, J. Park and K. Sekigawa were the first authors who defined the concept of a weakly Einstein Riemannian manifold as a modification of that of an Einstein Riemannian manifold. The defining formula is expressed in terms of the Riemannian scalar invariants of degree two. This concept was inspired by that of a super-Einstein manifold introduced earlier by A. Gray and T. J. Willmore in the context of mean-value theorems in Riemannian geometry. The dimension is the most interesting case, where...
The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition is said to be of type . The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers...
We discuss which semisimple locally symmetric spaces admit an AHS-structure invariant under local symmetries. We classify them for all types of AHS-structures and determine possible equivalence classes of such AHS-structures.