Nagoya Mathematical Journal

Geometric quantities of manifolds with Grassmann structure

N. Bokan, P. Matzeu, and Z. Rakić

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Abstract

We study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group $G = GL(p, \R) \otimes GL(q, \R)$ is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

Article information

Source
Nagoya Math. J. Volume 180 (2005), 45-76.

Dates
First available in Project Euclid: 14 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1134569895

Mathematical Reviews number (MathSciNet)
MR2186668

Zentralblatt MATH identifier
1093.53052

Subjects
Primary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}

Keywords
holonomy group Grassmann manifold normalization torsion-free connection action of a group irreducible representation

Citation

Bokan, N.; Matzeu, P.; Rakić, Z. Geometric quantities of manifolds with Grassmann structure. Nagoya Math. J. 180 (2005), 45--76.https://projecteuclid.org/euclid.nmj/1134569895


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