We prove that a Sasakian 3-manifold admitting a non-trivial solution to the Einstein-Dirac equation has necessarily constant scalar curvature. In the case when this scalar curvature is non-zero, their classi cation follows then from a result by Th. Friedrich and E.C. Kim. We also prove that a scalarat Sasakian 3-manifold admits no local Einstein spinors.

In this paper, we study gradient Ricci expanding solitons (X, g) satisfying Rc = cg +Df, where Rc is the Ricci curvature, c < 0 is a constant, and Df is the Hessian of the potential function f on X . We show that for a gradient expanding soliton (X, g) with non-negative Ricci curvature, the scalar curvature R has at least one maximum point on X , which is the only minimum point of the potential...

We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type infinite loop spaces. Our result in particular applies to path component round metric inside R+(Sd) if d≥6. To achieve goal, we study cobordism category manifolds with curvature. Under suitable connectivity conditions, can identify fiber forgetful map from psc ordinary a delooping metrics. This uses vers...

In this paper,we obtain two results on closed Reimainnian manifold M × [0, T ].When T is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When T is large and the given scalar curvature is small enough,the same result holds.

Inspired by the example of Abdelqader and Lake for the Kerr metric, we construct local scalar polynomial curvature invariants that vanish on the horizon of any stationary black hole: the squared norms of the wedge products of n linearly independent gradients of scalar polynomial curvature invariants, where n is the local cohomogeneity of the spacetime.

In this paper we study n-dimensional compact minimal submanifolds in S with scalar curvature S satisfying the pinching condition S > n(n − 2). We show that for p ≤ 2 these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension p ≥ 2, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. T...

Using Seiberg-Witten theory, it is shown that any Kähler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L-norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H(M) = H ⊕ H−. This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric.

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...