Positive curvature, symmetry, and topology

We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.

FUNDAMENTAL GROUPS OF CLOSED POSITIVELY CURVED MANIFOLDS WITH ALMOST DISCRETE ABELIAN GROUP ACTIONS

Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.

A COMPLETE METRIC OF POSITIVE CURVATURE ON R^n AND EXISTENCE OF CLOSED GEODESICS

An example of complete Riemannian metric of positive (or nonnagative) curvature on Resuch as aam = a(x)dx' is obtained by direct caculations. Furthermore, by using a geodesic convex condition and a theorem for complete noncompact Riemannian manifold, an existence resultof periodic solution of prescribed energy for a singular Handltonian system is also obtained.

Manifolds of positive Ricci curvature,quadratically asymptotically nonnegative curvature,and infinite Betti numbers

In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than or equal to 6.The purpose of the present paper is to use a different technique to exhibit a family of complete I-dimensional(I≥5)Riemannian manifolds of positive Ricci curvature,quadratically asymptotically nonnegative sectional curvature,and certain infinite Betti numbers bj(2≤j≤I-2).

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

We study the injectivity radius bound for 3-d Ricci flow with bounded curvature. As applications, we show the long time existence of the Ricci flow with positive Ricci curvature and with curvature decay condition at infinity. We partially settle a question of Chow-Lu-Ni [Hamilton＇s Ricci Flow, p. 302].

Four-manifolds with positive isotropic curvature

We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature （PIC）. Then we review an application of the classification theorem, which is from Chen and Zhu＇s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.

Metrics with Positive Scalar Curvature at Infinity and Localization Algebra

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.

Harmonic 2-forms and positively curved 4-manifolds

We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a strengthened Kato type inequality,then it is definite.We also discuss some new insights for compact Riemannian 4-manifolds with positive sectional curvature.

Spaces and moduli spaces of Riemannian metrics

These notes present and survey results about spaces and moduli spaces of complete Riemannian metrics with curvature bounds on open and closed manifolds, here focussing mainly on connectedness and disconnectedness properties. They also discuss several open problems and questions in the field.

On the Generalized Geroch Conjecture for Complete Spin Manifolds

Let W be a closed area enlargeable manifold in the sense of Gromov-Lawson and M be a noncompact spin manifold,the authors show that the connected sum M#W admits no complete metric of positive scalar curvature.When W=T^(n),this provides a positive answer to the generalized Geroch conjecture in the spin setting.

An isometrical CP^n-theorem

et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, then Mi is isometric to s^ni/Zh, CP^m/2, or CP^ni/2/Z2 with the canonical metric when ni 〉 0, and thus, M is isometric to Sn/Zh, CPn/2, or CPn/2/Z2 except possibly iso when n = 3 and M1 （or M2） ≌ S1/Zh with h ≥ 2 or n iso= 4 and M1 （or M2） iso ≌ RP^2