Normal holonomy in Lorentzian space and submanifold geometry

Abstract

We prove the polarity of the normal holonomy of riemannian submanifolds of lorentzian space. Using this result we prove that, essentially, there is no submanifold of hyperbolic space which admits a parallel normal field ξ ≠ 0 whose shape operator A ξ has constant eigenvalues. We prove the same result for submanifold of euclidean space by regarding them as submanifolds of a horosphere. This is by recovering the zero eigendistribution of A ξ by the normal holonomy of some riemannian submanifold of lorentzian space In particular, this implies that a homogeneous submanifold with parallel mean curvature H must be totally geodesic (the case H = 0 is a consequence of previous results of Di Scala, in the euclidean case, and Di Scala and the first author in the hyperbolic case). We also prove a generalization, using very simple and geometric facts, of the Homogeneous Slice Theorem of Heintze, Thorbergsson and the first author.

A Berger type normal holonomy theorem for complex submanifolds

“…A local group of isometries G of a Riemannian manifold X is said to act locally polarly if the distribution of normal spaces to maximal dimensional (local) orbits is integrable (or, equivalently, autoparallel; see [24]). If G acts locally polarly on X and S ⊂ X is a locally G-invariant submanifold, then the restriction of G to S acts locally polarly on S (this follows from Corollary 3.2.5 and Proposition 3.2 of [4], though we will only need the special cases given by Proposition 3.2.9 of this reference and Lemma 2.6 in [22]). …”

“…The Normal Holonomy Theorem was extended to Riemannian submanifolds of the Lorentz space [22]. The conclusion, in this case, is that the normal holonomy acts polarly on the (Lorentzian type) normal space.…”

“…In the Euclidean space this is well-known, since singular orbits of s-representation must have non-trivial normal holonomy [14]. In the Lorentzian space the polar actions are, orbit-like, essentially the same as in Euclidean space, up to some transitive factors in hyperbolic space or horospheres (see [22,Theorem 2.3]). …”

“…This construction is implicit in the proof of [22, Lemma 2.6]). The main point is that the restriction, to an invariant submanifold, of a locally polar action is again locally polar [22,Lemma 2.6]. Indeed,…”

“…From the Homogeneous Slice Theorem [13] (the local version follows from Theorem 3.1 in [22]) one has that, starting from x ∈ M and moving perpendicularly to E ξ 0 one can reach any other point of S(x). Indeed, let D = V ∩ E ξ 0 .…”