On the boundary behavior of left-invariant Hitchin and hypo flows

Abstract

We investigate left-invariant Hitchin and hypo flows on 5-, 6-and 7-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in SU(3), G 2 and Spin (7), respectively, which are in general geodesically incomplete. Generalizing results of Conti, we prove that for large classes of solvable Lie groups G these manifolds cannot be completed: a complete Riemannian manifold with parallel SU(3)-, G 2 -or Spin(7)-structure which is of cohomogeneity one with respect to G is flat, and has no singular orbits.We furthermore classify, on the non-compact Lie group SL(2, C), all half-flat SU(3)-structures which are bi-invariant with respect to the maximal compact subgroup SU(2) and solve the Hitchin flow for these initial values. It turns out that often the flow collapses to a smooth manifold in one direction. In this way we recover an incomplete cohomogeneity-one Riemannian metric with holonomy equal to G 2 on the twisted product SL(2, C) × SU(2) C 2 described by Bryant and Salamon. 1 Flow equations and special holonomyIn this section we give a brief overview of SU(2)-, SU(3)-, G 2 -and Spin (7)structures in dimension five, six, seven and eight, respectively, and their relation to the special holonomy groups SU(3), G 2 and Spin(7) in six, seven and eight dimensions, respectively, via certain flow equations. For more details on SU(3)-, G 2 -and Spin(7)-structures and proofs of the mentioned facts, the reader may consult, e.g., [10], [19] and [20]. For SU(2)-structures, the main references are [8] and [9]. Note that in [29] a unified treatment of all cases is given.We begin with the definition of the mentioned G-structures:Definition 2.1.• An SU(2)-structure on a five-dimensional manifold M is a quadruple (α, ω 1 , ω 2 , ω 3 ) ∈ Ω 1 M × (Ω 2 M ) 3 for which at each point p ∈ M there exists an ordered basis (e 1 , . . . , e 5 ) of T p M with α p = e 5 , (ω 1 ) p = e 12 + e 34 , (ω 2 ) p = e 13 − e 24 , (ω 3 ) p = e 14 + e 23 .The automorphism group of the above defined structure (i.e., the group of transformations preserving the forms α, ω 1 , ω 2 , ω 3 ) is SU(2) ⊂ SO(5). (α, ω 1 , ω 2 , ω 3 ) is called hypo if d ω 1 = 0, d(α ∧ ω 2 ) = 0, d(α ∧ ω 3 ) = 0.

Invariant torsion and G2-metrics

“…As we are free to rescale the metric on each summand V, this representation theoretical definition of an SO(3)-structure does not determine a canonical inclusion SO(3) ⊂ SO (6). We shall address this subtlety shortly.…”

“…The centralizer of SO(3) in GL(6, R) is a copy of GL(2, R) which contains the above rotation group SO (2). A first clear indication that this maximal commuting subgroup should not be ignored is the fact that it can be used to remedy the ambiguity in our choice of inclusion SO(3) ⊂ SO (6).…”

“…allows us to consider the module so(3) ⊥ which is the orthogonal complement of so(3) inside so (6). This latter module is reducible with irreducible summands given by R ⊕ 2V ⊕ S 2 0 (V).…”

“…Hence, only the expression (2.5) needs to be explained. The intrinsic torsion takes values in the cokernel of the alternating map (2.3) and this map is injective because of the inclusion so(3) ⊂ so (6). As SO(3) × SL(2, R)-modules, we therefore have that the intrinsic torsion belongs to the quotient…”

“…The Bryant-Salamon metric appears as a Σ 2 -invariant solution. On the open subset ∆ < 0, one can recover the metrics of [6].…”