2,266 research outputs found
Extending tensors on polar manifolds
Let be a Riemannian manifold with a polar action by the Lie group ,
with section and generalized Weyl group . We show that
restriction to is a surjective map from the set of smooth
-invariant tensors on onto the set of smooth -invariant tensors on
. Moreover, we show that every smooth -invariant Riemannian metric
on can be extended to a smooth -invariant Riemannian metric on
with respect to which the -action remains polar with the same section
.Comment: arXiv admin note: text overlap with arXiv:1205.476
Sectional curvature and Weitzenb\"ock formulae
We establish a new algebraic characterization of sectional curvature bounds
and using only curvature terms in the Weitzenb\"ock
formulae for symmetric -tensors. By introducing a symmetric analogue of the
Kulkarni-Nomizu product, we provide a simple formula for such curvature terms.
We also give an application of the Bochner technique to closed -manifolds
with indefinite intersection form and or , obtaining new
insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.
Strongly positive curvature
We begin a systematic study of a curvature condition (strongly positive
curvature) which lies strictly between positive curvature operator and positive
sectional curvature, and stems from the work of Thorpe in the 1970s. We prove
that this condition is preserved under Riemannian submersions and Cheeger
deformations, and that most compact homogeneous spaces with positive sectional
curvature satisfy it.Comment: LaTeX2e, 26 page
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