arXiv research

Study on curvature of cohomogeneity two Bazaikin spaces.

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension $…

Curvature study on Eschenburg spaces with two metrics.

Neural network improves normality testing accuracy.

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H som…

We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metr…

Proposes a method for valid inference in GPLSIMs with longitudinal data.

Classifies manifolds with quasipositive curvature.

The paper explores traveling along broken geodesics in Finsler submersions.

We formulate extensions of Wilking's Jacobi field splitting theorem to uniformly positive sectional curvature and also to positive and nonnegative intermediate Ricci curvatures.

This paper has been withdrawn by the author due to a critical error in the proof of Theorem A pointed out by Burkhard Wilking.

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wi…

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…

New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.

Generalizes symmetries of curved manifolds.

We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking's connectivity principle and Frankel's Theorem to the context of ${k}$-th Ricci curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the $C^{2}$-topology.

We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.

Study new Ricci flow invariant curvature conditions.

Proves conjecture about foliations on curved spaces.

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we con…

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely …

Complete Ricci flow from singular 3D manifold with pseudolocality.

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We…

For any complete $n$-dim Riemannian manifold $M^n$ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group $π_1(M^n)$ can be generated by $C(n)$ generators. Inspired by their work, we give a quantitative proof of the above theorem and show that $C(n)\…

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of certain cohomogeneity one manif…

Investigates second best Einstein manifolds in low dimensions.

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5…

B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$, which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature operators of a Kähler manifold of complex dimension $n$, one considers $Ad_{GL(n,\…

Note establishes a local maximum principle for Ricci flow under curvature conditions.

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Zille…

Ricci flow can change metrics with intermediate curvatures.

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive section…

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…

Logistic regression is used thousands of times a day to fit data, predict future outcomes, and assess the statistical significance of explanatory variables. When used for the purpose of statistical inference, logistic models produce p-values for the regression coefficients by using an approximation to the distribution …

Compact RCD(K,N) spaces with maximal rank are homeomorphic to infranilmanifolds.

The abstract discusses metrics with positive biorthogonal curvature on 5-manifolds.

Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative Ricci curvature, and if $n\geq 3$ we assume that $(M^n, g)\times \mathbb{R}$ has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on $(M^n, g)$ is proved. This provides an alternative proof for the uniform lower…

We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a $C^{1,α}$ compactness result for submanifolds, …

A compact Riemannian homogeneous space $G/H$, with a bi--invariant orthogonal decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in $T_{eH}(G/H)$ spanned by a linearly independent commuting pair in $\mathfrak{…

Improved theorem on curvature and manifold symmetry.

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Gr…

We show that the focal radius of any submanifold $N$ of positive dimension in a manifold $M$ with sectional curvature greater than or equal to $1$ does not exceed $\frac{π}{2}.$ In the case of equality, we show that $N$ is totally geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a projective s…

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly $\t…

We first notice in this article that if a compact Kähler manifold has the same integral cohomology ring and Pontrjagin classes as the complex projective space $\mathbb{C}P^n$, then it is biholomorphic to $\mathbb{C}P^n$ provided $n$ is odd. The same holds for even $n$ if we further assume that $M$ is simply-connected. …

This paper extends subordinated models to include stochastic time changes, improving financial modeling.

Improved bounds on $\Z_2$-torus actions on positively curved manifolds.

New framework improves classification accuracy using Pillai's trace and ULDA.

This article studies local and global inference for smoothing spline estimation in a unified asymptotic framework. We first introduce a new technical tool called functional Bahadur representation, which significantly generalizes the traditional Bahadur representation in parametric models, that is, Bahadur [Ann. Inst. S…