796 research outputs found
A note on the uniqueness of the canonical connection of a naturally reductive space
We extend the result in J. Reine Angew. Math. 664, 29-53, to the non-compact
case. Namely, we prove that the canonical connection on a simply connected and
irreducible naturally reductive space is unique, provided the space is not a
sphere, a compact Lie group with a bi-invariant metric or its symmetric dual.
In particular, the canonical connection is unique for the hyperbolic space when
the dimension is different from three. We also prove that the canonical
connection on the sphere is unique for the symmetric presentation. Finally, we
compute the full isometry group (connected component) of a compact and locally
irreducible naturally reductive space.Comment: 7 page
A geometric proof of the Karpelevich-Mostow's theorem
In this paper we give a geometric proof of the Karpelevich's theorem that
asserts that a semisimple Lie subgroup of isometries, of a symmetric space of
non compact type, has a totally geodesic orbit. In fact, this is equivalent to
a well-known result of Mostow about existence of compatible Cartan
decompositions
Corrigendum for "A geometric proof of the Karpelevich-Mostow theorem"
Corollary 2.3 in our paper "A geometric proof of the Karpelevich-Mostow
theorem", Bull. Lond. Math. Soc. 41 (2009), no. 4, 634-638, is false. Here we
give a counterexample and show how to avoid the use of this corollary to give a
simpler proof of Karpelevich-Mostow theorem. We also include a short discussion
of the original proof by Karpelevich
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