Obstructions to positive curvature and symmetry

We discuss new obstructions to positive sectional curvature and symmetry. The main result asserts that the index of the Dirac operator twisted with the tangent bundle vanishes on a 2-connected manifold of dimension ≠8 if the manifold admits a metric of positive sectional curvature and isometric effective S¹-action. The proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds.     

Parallel Codazzi tensors with submanifold applications

A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.     

KILLING VECTOR FIELDS OF CONSTANT LENGTH ON LOCALLY SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of the circle S ¹. Examples of unit Killing vector fields generated by almost free but not free actions of S ¹ on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated. DOI: .     

Rigidity of Einstein 4-manifolds with positive curvature

An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε₀, ε₀≡(-23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S ⁴, RP ⁴ with constant sectional curvature K=1/3, or CP ² with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S ⁴ and CP ² contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S ⁴, RP ⁴, or CP ² but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions. Oblatum 4-II-1999 & 4-V-2000?Published online: 16 August 2000     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Asymptotical flatness and cone structure at infinity

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends, and we classify (except for the case dim M=4, where it remains open if one of the theoretically possible cones can actually arise) for simply connected ends all possible cones at infinity. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat m-manifold with nonnegative sectional curvature is isometric to , whereS is an asymptotically flat surface. Received: 5 January 2000 / Published online: 19 October 2001     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Hyperfoliations on compact 3-manifolds with restrictions on the external curvature of leaves

C∞-foliations of codimension 1 on compact Riemannian 3-manifolds are studied. New classes of foliations, namely hyperbolic, elliptic, and parabolic foliations, are considered. Examples of such foliations are presented. In particular, aC∞-metric of nonnegative sectional curvature onS ³ such that the Reeb foliation is parabolic with respect to this metric is constructed. Analytic 3-manifolds with sectional curvature of constant sign admitting parabolic foliations are classified. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 651–659, May, 1998. The author wishes to express his thanks to Professor A. A. Borisenko for his supervision, and to Yu. A. Nikolaevskii for useful advice in the process of preparing the present paper.     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.     

Deformations of the hemisphere that increase scalar curvature

Consider a compact Riemannian manifold M of dimension n whose boundary ?M is totally geodesic and is isometric to the standard sphere S ^n?1. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n?1), then M is isometric to the hemisphere $S_{+}^{n}$ equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. In this paper, we construct counterexamples to Min-Oo??s Conjecture in dimension n??3.     

The sectional curvature remains positive when taking quotients by certain nonfree actions

We study some cases in which the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups S ¹ and S ³ for which the quotient space can be endowed with a smooth structure by means of the fibrations S ³/S ¹}~S ² and S ⁷/S ³}?S ⁴. We prove that the quotient space possesses a metric of positive sectional curvature provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.     

Islands at infinity on manifolds of asymptotically nonnegative curvature

We introduce an invariant which measures the R-eccentricity of a point in a complete Riemannian manifold M and show that it goes to zero when the point goes to infinity, if M has asymptotically nonnegative curvature. As a consequence we show that the isometry group is compact if M has asymptotically nonnegative curvature and a point with positive sectional curvature. Both authors were partially supported by CNPq of Brazil and the second author was also partially supported by FAPERJ of Brazil.     

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M ^G . We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M ^G . Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.     

Liouville theorems for exponentially harmonic functions on Riemannian manifolds

Suppose thatM is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that any bounded exponentially harmonic function onM is a constant function.     

Nonnegative sectional curvature hypersurfaces in a real space form

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form M ⁿ⁺¹(c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M ⁿ⁺¹(c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S ^k (a) × S ^n-k (√1 ? a ²), 1 ≤ k ≤ n ? 1, in S ⁿ⁺¹(1) and the Riemannian product H ^k (tanh² r ? 1) × S ^n-k (coth² r ? 1), 1 ≤ k ≤ n ? 1, in H ⁿ⁺¹(?1).     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.     

On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and L 2 harmonic forms

Let M ⁿ be a complete oriented noncompact hypersurface in a complete Riemannian manifold N ⁿ⁺¹ of nonnegative sectional curvature with ${2 \leq n \leq 5}$ . We prove that if M satisfies a stability condition, then there are no non-trivial L ² harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of ${\mathbb{R}}$ and a compact manifold with nonnegative sectional curvature. In case ${n \geq 5}$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.