In an article of Cao-Shen-Zhu [C-S-Z], they proved that a complete, immersed, stable minimal hypersurface M of R with n ≥ 3 must have only one end. When n = 2, it was proved independently by do Carmo-Peng [dC-P] and FischerColbrie-Schoen [FC-S] that a complete, immersed, oriented stable minimal surface in R must be a plane. Later Gulliver [G] and Fischer-Colbrie [FC] proved that if a complete, ...

1. Statement of results. In this paper we announce a substantial extension of the obstruction theory developed in [ l ] . Detailed proofs and further properties will appear elsewhere. Basically what we have accomplished is to put the theory in [l ] in a more natural setting and extend it to the metastable range. We also apply it to the question of when the Thorn complex JT(£) of a vector bundle...

In general, the converse of the above theorem does not hold. It requires a more refined invariant than the Lefschetz number to make the converse hold (see [1–3]). For this work, we focus on the similar arguments as above for the family of smooth maps over a compact base space B. The proof of the main theorem depends heavily on the intersection problem as follows. From now on, the notations X me...

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy th...

A Riemann-Roch theorem asserts that some algebraically defined wrong– way map in K-theory agrees with a topologically defined one [BFM]. Bismut and Lott [BiLo] proved a Riemann–Roch theorem for smooth fiber bundles in which the topologically defined wrong–way map is the homotopy transfer of Becker–Gottlieb and Dold. We generalize their theorem, refine it, and prove a converse stating that an ap...

T T T T T T T T T T T T T T T Abstract In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for mani-folds with compact boundary, the condition of inw...

A classical result of Toponogov [12] states that if a complete Riemannian manifold M with nonnegative sectional curvature contains a straight line, thenM is isometric to the metric product of a nonnegatively curved manifold and a line. We then know that the Busemann function associated with the straight line is an affine function, namely, a function which is affine on each unit speed geodesic i...

There are many previous finiteness theorems about diffeomorphism types in Riemannian geometry. Cheeger’s finiteness theorem asserts that given constants D, υ, and Λ, there are only finitely many n-dimensional compact differential manifoldX admitting Riemannian metric g such that diamg(X) 6 D, Volg(X) > υ and the sectional curvature |Sec(g)| 6 Λ. This theorem can be proved as a corollary of the ...