Related papers: A note on Canonical Ricci forms on 2-step nilmanif…

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic…

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.

We classify Einstein metrics on $\mathbb{R}^4$ invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. The metrics are either Ricci-flat or of negative Ricci curvature. We show that all…

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element \eta \in…

In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points…

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the…

We study Ricci-flat deformations of orbifolds in type II theory. We obtain a simple formula for mass corrections to the twisted modes due to the deformations, and apply it to originally tachyonic and massless states in several examples. In…

It is known that a closed collapsed Riemannian $n$-manifold $(M,g)$ of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of Cheeger-Fukaya-Gromov with respect to a smoothed metric…

The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the…

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with…

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex…

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. We completely compute these…

We present results indicating that the decomposition of a Ricci-flat manifold in its irreducible factors is reflected by the derived category of coherent sheaves. More precisely, we prove that a smooth projective variety that is derived…

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds…

For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants…

We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any…

We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such manifolds are conjectured to be K\"ahler (hence a complex space form) when the constant is non-zero and Chern flat (hence a quotient of…

Let $(M,g_0)$ be a compact $n$-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over $(\mathbb{S}^{n-1},g)$. We show that there exists a smooth Ricci…

Nilpotent Lie groups with left-invariant metrics provide non-trivial examples of Ricci solitons. One typical example is given by the class of two-step nilpotent Lie algebras obtained from simple directed graphs. In this paper, however, we…

We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive…