The index of symmetry of three-dimensional Lie groups with a left-invariant metric

Abstract

We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the socalled foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.

Homogeneous Riemannian Manifolds With Non-Trivial Nullity

“…If M is not locally symmetric, then its co-index of symmetry is at least 2. This was shown in [6] for the compact case and by Reggiani for the general case in [28,Thm. 2.2].…”

“…From the affine Killing equation and the Bianchi identity one can determine the initial conditions at p of the bracket [X, X ] of any two Killing fields in terms of the initial conditions (X) p = (v, B), (X ) p = (v , B ) (see [28,Lem. 2.4]).…”

“…In this section we construct examples of irreducible Riemannian homogeneous spaces with nullity of codimension 3 and co-index of symmetry 2 in any dimension greater than or equal to 4. As explained in the introduction such examples are optimal since neither the nullity can be greater nor the co-index of symmetry can be smaller due to Reggiani's Theorem [28].…”

Isometry groups of three-dimensional Lie groups

“…Recall that the index of symmetry is a geometric invariant, introduced in [ORT14], which measures how far is a homogeneous metric from being symmetric. The index of symmetry was successfully computed for several distinguished families of homogeneous spaces, such as compact naturally reductive spaces [ORT14] and naturally reductive nilpotent Lie groups [Reg19], flag manifolds [Pod15] and 3-dimensional unimodular Lie groups [Reg18]. It follows from our results that every 3-dimensional Lie group admits a left invariant metric with non-trivial index of symmetry.…”

“…The index of symmetry is a geometric invariant which measures how far is ( , ) from being a symmetric space, in the sense that ( , ) = dim if and only if ( , ) is a symmetric space. It is known that the distribution of symmetry can never have corank equal to 1 (see [Reg18]). In particular, if has dimension 3 and ( , ) is not a symmetric space, then the index of symmetry is equal to 0 or 1.…”