Metric Of Positive Sectional Curvature Research Articles

Infinite families of manifolds of positive kmathrm{th}-intermediate Ricci curvature with k small

Positive kmathrm{th}-intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by {{,mathrm{Ric},}}_k>0, is a condition that interpolates between positive sectional and positive Ricci curvature (when k =1 and k=n-1 respectively). In this work, we produce many examples of manifolds of {{,mathrm{Ric},}}_k>0 with k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension nge 7 congruent to 3,{{,mathrm{mod},}}4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of {{,mathrm{Ric},}}_k>0 for some k<n/2. We also prove that each dimension nge 4 congruent to 0 or 1,{{,mathrm{mod},}}4 supports closed manifolds which carry metrics of {{,mathrm{Ric},}}_k>0 with kle n/2, but do not admit metrics of positive sectional curvature.

Conformal Metrics to a Product or Doubly Warped Product on S 2 × S 2 and the Hopf Conjecture

Hopf’s well-known conjecture states that there exists no metric of positive sectional curvature in the product manifold S 2 × S 2 . In this paper, we show that the Hopf conjecture is true for conformal metrics to the product metric or doubly warped products on S 2 × S 2 .

AnHP2-bundle overS4with nontrivial Â-genus

We explain the existence of a smooth $HP^2$-bundle over $S^4$ whose total space has nontrivial $\hat{A}$-genus. Combined with an argument going back to Hitchin, this answers a question of Schick and implies that the space of Riemannian metrics of positive sectional curvature on a closed manifold can have nontrivial higher rational homotopy groups.

On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection

One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known.&#x0D; In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets

We construct non-trivial elements of order 2 in the homotopy groups $\pi_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $\Omega^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $\pi_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-invariant. As consequences, we (a) obtain non-trivial elements of order 2 in $\pi_* Diff(D^m,\partial)$ for m greater or equal to 6, and * + m congruent 0 or 1 modulo 8; (b) these elements remain non-trivial in $\pi_* Diff(M)$ where M is a closed spin manifold of the same dimension m and * > 0; (c) they act non-trivially on the corresponding homotopy group of the space of metrics of positive scalar curvature of such an M; in particular these homotopy groups are all non-trivial. The same applies to all other diffeomorphism invariant metrics of positive curvature, like the space of metrics of positive sectional curvature, or the space of metrics of positive Ricci curvature, provided they are non-empty. Further consequences are: (d) any closed spin manifold of dimension m greater or equal to 6 admits a metric with harmonic spinors; (e) there is no analogue of the odd-primary splitting of $(PL/O)_{(p)}$ for the prime 2; (f) for any $bP_{8j+4}$-sphere (where j > 0) of order which divides 4, the corresponding element in $\pi_0 Diff(D^{8j+2},\partial)$ lifts to $\pi_{8j-4} Diff(D^6,\partial)$, i.e., lies correspondingly deep down in the Gromoll filtration.

On a generalized conjecture of Hopf with symmetry

A famous conjecture of Hopf states that$\mathbb{S}^{2}\times \mathbb{S}^{2}$does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product$N\times N$can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.

Classifying seven dimensional manifolds of fixed cohomology type

We study seven dimensional manifolds of fixed cohomology type with integer coefficients: H0≅H2≅H5≅H7≅Z, H4≅Zr, H1=H3=H6=0, simply called manifolds of type r, where Zr, is a cyclic group of order r generated by the square of the generator of H2. Such manifolds include the Eschenburg spaces, the Witten manifolds and the generalized Witten manifolds. Most spaces from these three families admit a Riemannian metric of positive sectional curvature or an Einstein metric of positive Ricci curvature. In 1991 M. Kreck and S. Stolz introduced three invariants to classify manifolds M of type r up to homeomorphism and diffeomorphism. In this article, we show that for spin manifolds of type r we can replace two of the homeomorphism invariants by the first Pontrjagin class and the self-linking number of the manifolds. This replacement has been stated without proof and been used for calculations for the Eschenburg spaces, hence our result closes a gap in the literature.

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO ( 3 ) -bundle is diffeomorphic to either S 4 or C P ¯ 2 . The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S 4 and C P ¯ 2 are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang–Kleiner's theorem that only S 4 and C P 2 admit circle-invariant metrics of positive sectional curvature.

Positively Curved GKM-Manifolds

Let T be a torus of dimension at least k and M a T-manifold. M is a GKM_k-manifold if the action is equivariantly formal, has only isolated fixed points, and any k weights of the isotropy representation in the fixed points are linearly independent. In this paper we compute the cohomology rings with real and integer coefficients of GKM_3- and GKM_4-manifolds which admit invariant metrics of positive sectional curvature.

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.

Einstein Hermitian metrics of positive sectional curvature

It is shown that, up to scaling and isometry, the only complete 4-manifold with an Einstein metric of positive sectional curvature which is also Hermitian with respect to some complex structure is the complex projective plane C P 2 \mathbb {CP}_2 , equipped with its Fubini-Study metric.

A 7-manifold with positive curvature

We demonstrate the existence of a metric of positive sectional curvature on the 3-Sasakian 7-manifold over Hitchin's self-dual Einstein orbifold with k=5.

Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

The authors give a short survey of previous results on generalized normal homogeneous ( δ-homogeneous, in other terms) Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with nonnegative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable generalized normal homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds Sp ( l ) / U ( 1 ) ⋅ Sp ( l − 1 ) = C P 2 l − 1 , l ⩾ 2 , supplied with invariant Riemannian metrics of positive sectional curvature with the pinching constants (the ratio of the minimal sectional curvature to the maximal one) in the open interval ( 1 / 16 , 1 / 4 ) . This implies very unusual geometric properties of the adjoint representation of Sp ( l ) , l ⩾ 2 . Some unsolved questions are suggested.

Topological structure of candidates for positive curvature

In light of recent advances in the study of manifolds admitting Riemannian metrics of positive sectional curvature, the study of certain infinite families of seven dimensional manifolds has become a matter of interest. We determine the cohomology ring structures of manifolds belonging to these families. This particular ring structure indicates the existence of topological invariants distinguishing the corresponding homeomorphism and diffeomorphism type. We show that all families contain representatives of infinitely many homotopy types.

Positively Curved Self-Dual Einstein Metrics on Weighted Projective Planes

We classify positively curved self-dual Einstein Hermitian orbifold metrics of Galicki – Lawson on the weighted projective planes. We thus determine which of the 3-Sasakian S1-reductions of S11 possess canonical variation metrics of positive sectional curvature.

Isometry groups of homogeneous spaces with positive sectional curvature

We calculate the full isometry group in the case G/ H admits a homogeneous metric of positive sectional curvature.

A condition for positivity of curvature

In this note we describe a necessary and sufficient condition, in order that a procedure of the type described in [D-R] yields metrics of positive sectional curvature in the total space of a principal bundle.

Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature

Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature

Equivariant geometry and Kervaire spheres

The intrinsic geometry of metrics on the Kervaire sphere which are invariant under a large transformation group (cohomogeneity one) is studied. Invariant theory is used to describe the behavior of these metrics near the singular orbits. Nice expressions for the Ricci and sectional curvatures are obtained. The nonexistence of invariant metrics of positive sectional curvature is proven, and Cheeger’s construction of metrics of positive Ricci curvature is discussed.

Unflat connections in 3-sphere bundles over 𝑆⁴

The paper concerns connections in 3 3 -sphere bundles over 4 4 -manifolds having the property of unflatness, which is a necessary condition in order that a natural construction give a Riemannian metric of positive sectional curvature in the total space. It is shown that, as conjectured by A. Weinstein, the only 3 3 -sphere bundle over S 4 {S^4} with an unflat connection is the Hopf bundle.