On the boundary behavior of left-invariant Hitchin and hypo flows
نویسندگان
چکیده
We investigate left-invariant Hitchin and hypo flows on 5-, 6and 7-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in SU(3), G2 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)-, G2or 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 G2 on the twisted product SL(2,C)×SU(2) C 2 described by Bryant and Salamon.
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عنوان ژورنال:
- J. London Math. Society
دوره 92 شماره
صفحات -
تاریخ انتشار 2015