Related papers: On conformally flat manifolds with constant positi…

We prove that compact non-flat manifolds with constant sectional curvature admit no conformal product structure. Furthermore, we demonstrate that the methods extend naturally to irreducible, compact locally symmetric spaces of non-positive…

We show the contractibility of spaces of invariant Riemannian metrics of positive scalar curvature on compact connected manifolds of dimension at least two, with and without boundary and equipped with compact Lie group actions. On manifolds…

We prove that a closed $n$-manifold $M$ with positive scalar curvature and abelian fundamental group admits a finite covering $M'$ which is strongly inessential. The latter means that a classifying map $u:M'\to K(\pi_1(M'),1)$ can be…

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{2}{3}$, then it decomposes as a (possibly infinite)…

We classify conformally flat Riemannian $3-$manifolds which possesses a free isometric $S^1-$action.

We prove that the positive mass theorem applies to Lipschitz metrics as long as the singular set is low-dimensional, with no other conditions on the singular set. More precisely, let $g$ be an asymptotically flat Lipschitz metric on a…

We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism…

we show that the space of metrics of positive scalar curvature on a manifold is, when nonempty, homotopy equivalent to a space of metrics of positive scalar curvature that restrict to a fixed metric near a given submanifold of codimension…

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes…

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.

We study the classification of closed, smooth, spin, $1$-connected $7$-manifolds whose integral cohomology ring is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology ring of a closed, smooth, spin,…

On a manifold $(\mathbb{R}^n, e^{2u} |dx|^2)$, we say $u$ is normal if the $Q$-curvature equation that $u$ satisfies $(-\Delta)^{\frac{n}{2}} u = Q_g e^{nu}$ can be written as the integral form $u(x)=\frac{1}{c_n}\int_{\mathbb…

An $n$-dimensional ($n\geq 2$) simply connected, compact without boundary Finsler space of positive constant sectional curvature is conformally homeomorphic to an n-sphere in the Euclidean space $\R^{n+1}$.

In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity…

In 2014, Gromov asked if nonnegative scalar curvature is preserved under intrinsic flat convergence. Here we construct a sequence of closed oriented Riemannian $n$-manifolds, $n\geq 3$, with positive scalar curvature such that their…

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar…

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a…

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar…

In this paper, we give various curvature pinching conditions such that shrinkers are compact. On one hand, we prove that shrinkers with positive Ricci curvature are compact when they have bounded curvature and certain curvature pinching…

The Schouten tensor \ $A$ \ of a Riemannian manifold \ $(M,g)$ provides important scalar curvature invariants $\sigma_k$, that are the symmetric functions on the eigenvalues of $A$, where, in particular, $\sigma_1$ \ coincides with the…