We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e., where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e., the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and at invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent.
HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY / Di Scala, A. J.; Olmos, C.; Vittone, F.. - In: TRANSFORMATION GROUPS. - ISSN 1083-4362. - ELETTRONICO. - (2020). [10.1007/s00031-020-09611-2]
Titolo: | HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY | |
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Data di pubblicazione: | 2020 | |
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Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s00031-020-09611-2 | |
Appare nelle tipologie: | 1.1 Articolo in rivista |
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http://hdl.handle.net/11583/2922700