Spaces of positive intermediate curvature metrics

Geometriae Dedicata - In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive...     

Finsler metrics with positive constant flag curvature

We showed that any reversible Finsler metric with positive constant flag curvature must be Riemannian. Received: 18 August 2008     

The space of metrics of positive scalar curvature

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics. Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing \(\hat{A}\) -genera, thus establishing the non-multiplicativity of the \(\hat{A}\) -genus in fiber bundles with simply connected base.     

Complete intersections with metrics of positive scalar curvature

We give a complete list of complex projective complete intersections admitting Riemannian metrics of positive scalar curvature. To cite this article: F. Fang, P. Shao, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Kähler-Einstein metrics with positive scalar curvature

In this paper, we prove that the existence of K?hler-Einstein metrics implies the stability of the underlying K?hler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact K?hler manifold admits K?hler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of K?hler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable K?hler manifolds, provided that the partial C ⁰-estimate posed in [T6] is true. Oblatum 12-IV-1996 & 8-XI-1996     

A quasifibration of spaces of positive scalar curvature metrics

In this paper we show that for Riemannian manifolds with boundary the natural restriction map is a quasifibration between spaces of metrics of positive scalar curvature. We apply this result to study homotopy properties of spaces of such metrics on manifolds with boundary.

     

Invariant metrics of positive Ricci curvature on principal bundles

Let be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let be a principal bundle over with compact connected structure group . If the fundamental group of is finite, we show that admits a invariant metric with positive Ricci curvature so that is a Riemannian submersion. Received 14 January 1997     

The eta invariant and metrics of positive scalar curvature

Research partially supported by the NSF     

Some sufficient conditions for path-factor uniform graphs

Aequationes mathematicae - For a set $$mathcal {H}$$ of connected graphs, a spanning subgraph H of G is called an $$mathcal {H}$$ -factor of G if each component of H is isomorphic to an element...     

Some sufficient conditions for the observability of nonlinear systems

Some sufficient conditions are presented for the observability of systems described by nonlinear, ordinary differential equations with linear observations. The point which the authors especially emphasize is the fact that the results obtained are global in some sense. In the beginning, the observability problem is reduced to the uniqueness problem of solutions of some nonlinear integral equations for general nonlinear systems with linear observations. Then, for some restricted systems, a simple sufficient condition is derived. The relation between global and local observability for nonlinear systems is also considered.The authors wish to thank their colleagues in Nagoya University.     

On sufficient conditions for the positive solvability of the Riemann-Hilbert problem

Translated from Matematicheskie Zametki, Vol. 51, No. 2, pp. 9–19, February, 1992.     

Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of D is an independent set of vertices S of D such that for every z∈V(D)−S for which there exists an (S,z)-arc of D−F, there also exists an (z,S)-arc in D. In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F. As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.     

Riemannian metrics of positive scalar curvature on compact manifolds with boundary

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S ⁿ is a group.     

Some remarks on sufficient conditions for nonsmooth functions

We discuss some results of the paper Recht (1993) in which the author states first-order sufficient conditions for a local minimum in a nonsmooth setting using some kind of generalized derivatives. It will be shown that part of the results has to be improved. Moreover, it is shown that an extension to infinite-dimensional spaces is impossible.     

Some sufficient conditions for the existence of a 1-factor

The following theorem is proved: Let G be a graph of even order. Assume that there exists a connected spanning subgraph F of G such that for every set U of four vertices in G, if the subgraph of F induced by U is a star, then the subgraph of G induced by U is complete. Then G has a 1-factor. The above theorem is derived from another sufficient condition for the existence of a 1-factor, which is also proved in this paper (Lemma 1).     

Some necessary and sufficient conditions for Hypercyclicity Criterion

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion. This paper is a part of the second author’s Doctoral thesis, written at Shiraz University under the direction of the first author.