102 research outputs found
Evolution of the Weyl Tensor under the Ricci Flow
We compute the evolution equation of the Weyl tensor under the Ricci flow of
a Riemannian manifold and we discuss some consequences for the classification
of locally conformally flat Ricci solitons
Locally conformally flat ancient Ricci flows
We show that any locally conformally flat ancient solution to the Ricci flow
must be rotationally symmetric. As a by-product, we prove that any locally
conformally flat Ricci soliton is a gradient soliton in the shrinking and
steady cases as well as in the expanding case, provided the soliton has
nonnegative curvature.Comment: Final version, to appear on Anal. PD
A note on Codazzi tensors
We discuss a gap in Besse's book, recently pointed out by Merton, which
concerns the classification of Riemannian manifolds admitting a Codazzi tensors
with exactly two distinct eigenvalues. For such manifolds, we prove a structure
theorem, without adding extra hypotheses and then we conclude with some
application of this theory to the classification of three-dimensional gradient
Ricci solitons.Comment: Minor correction
On the Distributional Hessian of the Distance Function
We describe the precise structure of the distributional Hessian of the
distance function from a point of a Riemannian manifold. In doing this we also
discuss some geometrical properties of the cutlocus of a point and we compare
some different weak notions of Hessian and Laplacian
On the global structure of conformal gradient solitons with nonnegative Ricci tensor
In this paper we prove that any complete conformal gradient soliton with
nonnegative Ricci tensor is either isometric to a direct product
, or globally conformally equivalent to the Euclidean
space or to the round sphere . In particular,
we show that any complete, noncompact, gradient Yamabe-type soliton with
positive Ricci tensor is rotationally symmetric, whenever the potential
function is nonconstant.Comment: Minor correction
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