HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY
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Springer Science+Business Media New York (2020)
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HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY A. J. DI SCALA∗
C. OLMOS∗∗
Dipartimento di Scienze Matematiche Politecnico di Torino Corso Duca degli Abruzzi, 24 10129, Torino, Italy
FaMAF, CIEM-CONICET Universidad Nacional de C´ordoba Ciudad Universitaria 5000, C´ordoba, Argentina [email protected]
[email protected]
F. VITTONE∗∗∗ FCEIA-CONICET Universidad Nacional de Rosario Av. Pellegrini 250 2000, Rosario, Argentina [email protected]
Abstract. 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 flat 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. DOI: 10.1007/S00031-020-09611-2 Member of GNSAGA of INdAM and of DISMA Dipartimento di Eccellenza MIUR 2018-2022. ∗∗ Supported by FaMAF and CIEM-CONICET. ∗∗∗ Supported by FCEIA and CONICET. Received May 9, 2019. Accepted April 29, 2020. Corresponding Author: C. Olmos, e-mail: [email protected] ∗
A. J. DI SCALA, C. OLMOS, F. VITTONE
1. Introduction Given a Riemannian manifold M with curvature tensor R and a point p ∈ M , the nullity subspace νp of M at p is defined as the subset of Tp M consisting of those vectors that annihilate R, i.e., νp = {v ∈ Tp M : R ·, · v ≡ 0}. The concept of nullity of the curvature tensor was first introduced by Chern and Kuiper in [10]. For a general Riemannian manifold, the dimension of the nullity subspace at a point p, called the index of nullity at p, might change from point to point. In the open and dense subset Ω of M where the index of nullity is locally constant, q 7→ νq is an autoparallel distribution with flat (totally geodesic) integral manifolds. Moreover, in the open subset of Ω where the index of nullity attains its minimum, the integral m
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