Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

Harvey, John; Searle, Catherine

doi:10.1007/s12220-016-9734-7

Cited by 41 publications

(85 citation statements)

References 43 publications

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“…In particular, M//G has no 1-codimensional strata. The union of the closures of the 1-codimensional strata coincides with the Alexandrov boundary of M/G, so it follows that M//G is closed in the sense of [23].…”

Section: Positively Curved Foliations With Maximal-dimensional Closurementioning

confidence: 97%

“…The proof is similar to that of the classical case [13, Theorem 1] if one works with H Fgeodesics on a total transversal T F , so we will omit it. [23,Section 6]. By [23,Theorem 6.5], there is a unique orbit S 1 x at maximal distance from N, the "soul" orbit, and there is an S 1 -equivariant homeomorphism…”

Section: Transverse Hopf Conjecturementioning

confidence: 99%

“…[23,Section 6]. By [23,Theorem 6.5], there is a unique orbit S 1 x at maximal distance from N, the "soul" orbit, and there is an S 1 -equivariant homeomorphism…”

Section: Transverse Hopf Conjecturementioning

confidence: 99%

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Positively curved Killing foliations via deformations

Caramello

Töben

2019

Trans. Amer. Math. Soc.

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We show that a compact manifold admitting a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces, provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact we prove that a Riemannian foliation of a compact manifold with finite fundamental group and nonvanishing Euler characteristic is closed. As another application we obtain that, for a positively curved Killing foliation of a compact manifold, if the structural algebra has sufficiently large dimension then the basic Euler characteristic is positive.

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Section: Positively Curved Foliations With Maximal-dimensional Closurementioning

confidence: 97%

Section: Transverse Hopf Conjecturementioning

confidence: 99%

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Positively curved Killing foliations via deformations

Caramello

Töben

2019

Trans. Amer. Math. Soc.

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“…If X * is not orientable then we only consider the case when X is a boundaryless Alexandrov space. There is a notion of an orientation cover X of X [16]. It is also an Alexandrov space and is equipped with a Z 2 -action whose quotient is X.…”

Section: Differential Form Laplacian On An Alexandrov Spacementioning

confidence: 99%

Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

Lott

2018

Math. Ann.

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We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.

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“…In light of the fact that every object in this collection is the limit of smooth flat manifolds, it would be interesting to determine whether every orbifold with sec ≥ 0 is the limit of manifolds with sec ≥ 0; see Remark 2.1. An important and currently open question is whether every finite-dimensional Alexandrov space with curv ≥ K is the limit of smooth manifolds with sec ≥ K. In this context, recall that Alexandrov spaces of dimension 3 and 4 are homeomorphic to orbifolds [18,22].…”

Section: Introductionmentioning

confidence: 99%

Teichmüller theory and collapse of flat manifolds

Bettiol

Derdziński

Piccione

2018

Annali di Matematica

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We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by collapsing closed flat manifolds, and the collapsed limits of closed flat 3-manifolds are classified.representation, and we write K i for R, C, or H, according to this irreducible being of real, complex, or quaternionic type. The Teichmüller space T flat (O) is diffeomorphic to:where GL(m, K) is the group of K-linear automorphisms of K m and O(m, K) stands for O(m), U(m), or Sp(m), when K is, respectively, R, C, or H. In particular, T flat (O) is real-analytic and diffeomorphic to R d .which are given by:In particular, since the holonomy representation of a flat manifold is reducible [23], see Theorem 2.4, it follows that l ≥ 2, and hence d ≥ 2. This implies the following:Corollary C. Every flat manifold admits nonhomothetic flat deformations.The situation is different for flat orbifolds, which can be rigid. Examples of orbifolds with irreducible holonomy representation, i.e., l = 1, which consequently admit no nonhomothetic flat deformations, already appear in dimension 2: for instance, flat equilateral triangles; see Subsection 5.3 for more examples.Since flat orbifolds are locally isometric to Euclidean spaces, the most interesting aspects of their geometry are clearly global. Thus, it is not surprising that issues related to holonomy play a central role in developing this Teichmüller theory. As an elementary case illustrating Theorem B, consider the complete absence of holonomy: flat n-dimensional tori T n can be realized as parallelepipeds spanned by linearly independent vectors v 1 , . . . , v n ∈ R n , with opposite faces identified. Flat metrics on T n correspond to different choices of v 1 , . . . , v n , up to ambiguities arising from rigid motions in R n , or relabelings and subdivisions of the parallelepiped into smaller pieces with boundary identifications. More precisely, it is not difficult to see that M flat (T n ) = O(n)\GL(n, R)/GL(n, Z). In this case, T flat (T n ) = O(n)\GL(n, R) ∼ = R n(n+1)/2 is the space of inner products on R n , and M flat (T n ) = T flat (T n )/GL(n, Z); see Subsection 5.1 for details.Isometry classes of collapsed limits of T n correspond to points in the ideal boundary of M flat (T n ). A more tangible object is the ideal boundary of T flat (T n ), formed by positive-semidefinite n×n matrices and stratified by their rank k, with 0 ≤ k < n, which in a sense correspond to the k-dimensional flat tori T k to which T n can collapse. Nevertheless, we warn the reader that the Gromov-Hausdorff distance does not extend continuously to this boundary. For instance, collapsing the 2-dimensional square torus along a line of slope p/q, with p, q ∈ Z, gcd(p, q) = 1, produces as Gromov-Hausdorff limit the circle S 1 of length (p 2 + q 2 ) −1/2 , while collapsing it along any...

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Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

Cited by 41 publications

References 43 publications

Positively curved Killing foliations via deformations

Positively curved Killing foliations via deformations

Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

Teichmüller theory and collapse of flat manifolds

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