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Classification of (1,1) tensor fields and bihamiltonian structures

Francisco Turiel (1996)

Banach Center Publications

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 f 1 , . . . , f m , defined around p, such that ( d f 1 ∧ . . . ∧ d f m ) ( p ) ≠ 0 and d ( d f j ( J ( ) ) ) ( p ) = 0 , 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.

Classification of 4 -dimensional homogeneous weakly Einstein manifolds

Teresa Arias-Marco, Oldřich Kowalski (2015)

Czechoslovak Mathematical Journal

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 4 is the most interesting case, where...

Classification of 4-dimensional homogeneous D'Atri spaces

Teresa Arias-Marco, Oldřich Kowalski (2008)

Czechoslovak Mathematical Journal

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 ( M , g ) satisfying the first odd Ledger condition is said to be of type 𝒜 . 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...

Classification of principal connections naturally induced on W 2 P E

Jan Vondra (2008)

Archivum Mathematicum

We consider a vector bundle E → M and the principal bundle P E of frames of E . Let K be a principal connection on P E and let Λ be a linear connection on M . We classify all principal connections on W 2 P E = P 2 M × M J 2 P E naturally given by K and Λ .

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