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Calculus on symplectic manifolds

Michael Eastwood, Jan Slovák (2018)

Archivum Mathematicum

On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini–Study form and connection, we can build a series of differential complexes akin to the Bernstein–Gelfand–Gelfand complexes from parabolic differential geometry.

Characterization of totally umbilic hypersurfaces in a space form by circles

Toshiaki Adachi, Sadahiro Maeda (2005)

Czechoslovak Mathematical Journal

In this paper we characterize totally umbilic hypersurfaces in a space form by a property of the extrinsic shape of circles on hypersurfaces. This characterization corresponds to characterizations of isoparametric hypersurfaces in a space form by properties of the extrinsic shape of geodesics due to Kimura-Maeda.

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...

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