A rigidity theorem for manifolds without conjugate points

1998 ◽  
Vol 18 (4) ◽  
pp. 813-829 ◽  
Author(s):  
CHRISTOPHER B. CROKE ◽  
BRUCE KLEINER
Keyword(s):  
Companion Paper ◽  
Rigidity Theorem ◽  
Conjugate Points ◽  
Compactly Supported ◽  
Connected Case ◽  
Flat Metric ◽  
Simply Connected

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

A rigidity theorem for simply connected manifolds without conjugate points

1998 ◽  
Vol 18 (4) ◽  
pp. 807-812 ◽  
Author(s):  
CHRISTOPHER B. CROKE ◽  
BRUCE KLEINER
Keyword(s):  
Riemannian Manifold ◽  
Rigidity Theorem ◽  
Conjugate Points ◽  
Compact Set ◽  
Riemannian Product ◽  
Simply Connected Manifolds ◽  
Simply Connected

In this paper we show that manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, Section I.5]. The main theorem is that if $X$ is a complete, simply connected Riemannian manifold without conjugate points, and $M=X\times R$ is given the Riemannian product metric $g$, then any metric without conjugate points on $M$ which agrees with $g$ outside a compact set is isometric to $g$.

Minimum and Conjugate Points in Symmetric Spaces

1962 ◽  
Vol 14 ◽  
pp. 320-328 ◽  
Author(s):  
Richard Crittenden
Keyword(s):  
Symmetric Space ◽  
Algebraic Structure ◽  
Tangent Bundle ◽  
Tangent Space ◽  
Symmetric Spaces ◽  
Conjugate Points ◽  
Minimum Cut ◽  
Cut Locus ◽  
Connected Case ◽  
Simply Connected

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

Non-cocompact Group Actions and -Semistability at Infinity

2019 ◽  
Vol 72 (5) ◽  
pp. 1275-1303 ◽  
Author(s):  
Ross Geoghegan ◽  
Craig Guilbault ◽  
Michael Mihalik
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Fundamental Property ◽  
Companion Paper ◽  
Infinite Index ◽  
Locally Compact ◽  
Finitely Presented Groups ◽  
Simply Connected ◽  
Open Question ◽  
Finitely Presented

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

scholarly journals SPIKES IN COSMIC CRYSTALLOGRAPHY

2002 ◽  
Vol 11 (06) ◽  
pp. 869-891 ◽  
Author(s):  
G. I. GOMERO ◽  
A. F. F. TEIXEIRA ◽  
M. J. REBOUÇCAS ◽  
A. BERNUI
Keyword(s):  
Multiple Images ◽  
Multiply Connected ◽  
Major Consequence ◽  
Statistical Fluctuations ◽  
Covering Group ◽  
Connected Case ◽  
Pair Separation ◽  
Simply Connected ◽  
The Universe ◽  
Covering Groups

If the universe is multiply connected and small the sky shows multiple images of cosmic objects, correlated by the covering group of the 3-manifold used to model it. These correlations were originally thought to manifest as spikes in pair separation histograms (PSH) built from suitable catalogues. Using probability theory we derive an expression for the expected pair separation histogram (EPSH) in a rather general topological-geometrical-observational setting. As a major consequence we show that the spikes of topological origin in PSH's are due to translations, whereas other isometries manifest as tiny deformations of the PSH corresponding to the simply connected case. This result holds for all Robertson–Walker spacetimes and gives rise to two basic corollaries: (i) that PSH's of Euclidean manifolds that have the same translations in their covering groups exhibit identical spike spectra of topological origin, making clear that even if the universe is flat the topological spikes alone are not sufficient for determining its topology; and (ii) that PSH's of hyperbolic 3-manifolds exhibit no spikes of topological origin. These corollaries ensure that cosmic crystallography, as originally formulated, is not a conclusive method for unveiling the shape of the universe. We also present a method that reduces the statistical fluctuations in PSH's built from simulated catalogues.

scholarly journals Fourier transforms of powers of well-behaved 2D real analytic functions

2018 ◽  
Vol 30 (3) ◽  
pp. 723-732
Author(s):  
Michael Greenblatt
Keyword(s):  
Analytic Functions ◽  
Fourier Transforms ◽  
Newton Polygon ◽  
Companion Paper ◽  
Two Dimensional ◽  
Compactly Supported ◽  
Sharp Estimates ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Compactly Supported Functions

AbstractThis paper is a companion paper to [6], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [6] are stated in a rather general form. In this paper, we expand on the results of [6] and show that there is a class of “well-behaved” functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.

The Robin function and conformal welding – A new proof of the existence

2020 ◽  
Vol 27 (1) ◽  
pp. 43-51
Author(s):  
Bodo Dittmar
Keyword(s):  
Conformal Mapping ◽  
Boundary Value Problem ◽  
Harmonic Functions ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Uniformization Theorem ◽  
Robin Function ◽  
Connected Case ◽  
Simply Connected ◽  
Mathematical Physicist

AbstractGreen’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

scholarly journals AN INTRINSIC PROOF OF GROMOLL–GROVE DIAMETER RIGIDITY THEOREM

2007 ◽  
Vol 09 (03) ◽  
pp. 401-419 ◽  
Author(s):  
JIANGUO CAO ◽  
HONGYAN TANG
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Projective Spaces ◽  
Rigidity Theorem ◽  
Simply Connected

Using the spherical trip theorem, we present a new intrinsic proof of Gromoll–Grove diameter rigidity theorem: "If a simply-connected Riemannian manifold has sectional curvature ≥ 1 and diameter [Formula: see text], then either it is homeomorphic to a sphere, or it is isometric to one of classic projective spaces".

Surjectivity of the Taylor map for complex nilpotent Lie groups

2009 ◽  
Vol 146 (1) ◽  
pp. 177-195 ◽  
Author(s):  
BRUCE K. DRIVER ◽  
LEONARD GROSS ◽  
LAURENT SALOFF-COSTE
Keyword(s):  
Lie Algebra ◽  
Hermitian Form ◽  
Holomorphic Functions ◽  
Companion Paper ◽  
Wigner Transform ◽  
Enveloping Algebra ◽  
Taylor Map ◽  
Hörmander’S Condition ◽  
Simply Connected ◽  
Insight Into

AbstractA Hermitian formqon the dual space,*, of the Lie algebra,, of a simply connected complex Lie group,G, determines a sub-Laplacian, Δ, onG. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, onGassociated toetΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions inL2(G, ρt) ontoJt0(a subspace of) the dual of the universal enveloping algebra of. Here we give a very different proof of the surjectivity of the Taylor map under the assumption thatGis nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense inJt0when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions inL2(G, ρt), for appropriatet, whenGis the complex Heisenberg group.

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