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.
