scholarly journals Balanced presentations of the trivial group and four-dimensional geometry

2017 ◽  
Vol 09 (01) ◽  
pp. 15-25 ◽  
Author(s):  
Boris Lishak ◽  
Alexander Nabutovsky
Keyword(s):  
Smooth Manifold ◽  
Riemannian Metrics ◽  
Injectivity Radius ◽  
Trivial Group ◽  
Codimension One ◽  
Positive Formula

We prove that: (1) There exist infinitely many nontrivial codimension one “thick” knots in [Formula: see text]; (2) For each closed four-dimensional smooth manifold [Formula: see text] and for each sufficiently small positive [Formula: see text] the set of isometry classes of Riemannian metrics with volume equal to [Formula: see text] and injectivity radius greater than [Formula: see text] is disconnected; and (3) For each closed four-dimensional [Formula: see text]-manifold [Formula: see text] and any [Formula: see text] there exist arbitrarily large values of [Formula: see text] such that some two triangulations of [Formula: see text] with [Formula: see text] simplices cannot be connected by any sequence of [Formula: see text] bistellar transformations, where [Formula: see text] ([Formula: see text] times).

Harmonic functions on ℝ-covered foliations

2009 ◽  
Vol 29 (4) ◽  
pp. 1141-1161
Author(s):  
S. FENLEY ◽  
R. FERES ◽  
K. PARWANI
Keyword(s):  
Harmonic Functions ◽  
Riemannian Metrics ◽  
Related Result ◽  
Continuous Functions ◽  
Property A ◽  
Liouville Property ◽  
Foliated Manifold ◽  
Curved Manifolds ◽  
Codimension One ◽  
Negatively Curved

AbstractLet (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

Transverse, type changing, pseudo riemannian metrics and the extendability of geodesics

1994 ◽  
Vol 444 (1921) ◽  
pp. 297-306 ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Smooth Manifold ◽  
Local Existence ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Type I ◽  
Partial Differential ◽  
Geometric Conditions ◽  
Type Change

We study the geodesics and pre-geodesics of a smooth manifold with smooth pseudo riemannian metric which changes bilinear type (i. e. the signature changes) on a hypersurface. We classify all geodesics and pre-geodesics that cross the hypersurface of type change transversely. We then apply these results to the eikonal partial differential equation to find geometric conditions for the local existence or non-existence of smooth, transverse solutions.

scholarly journals Dirac and Plateau billiards in domains with corners

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Misha Gromov
Keyword(s):  
Dirac Operator ◽  
Riemannian Manifolds ◽  
Smooth Manifold ◽  
Riemannian Metrics ◽  
Continuous Functions ◽  
Plateau Problem ◽  
Singular Spaces ◽  
Topology Of Manifolds ◽  
Manifolds With Corners ◽  
Domains With Corners

AbstractGroping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.

scholarly journals Heterogeneous Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
James J. Hebda
Keyword(s):  
Riemannian Manifolds ◽  
Smooth Manifold ◽  
Riemannian Metrics ◽  
Generic Class

We solve Ambrose's Problem for a generic class of Riemannian metrics on a smooth manifold, namely, the class of heterogeneous metrics.

scholarly journals Existence of regular coverings associated with leaves of codimension one foliations

1977 ◽  
Vol 67 ◽  
pp. 15-34 ◽  
Author(s):  
Gikō Ikegami
Keyword(s):  
Line Bundle ◽  
Smooth Manifold ◽  
Riemannian Metric ◽  
Normal Line ◽  
Codimension One

In this paper we are concerned with transversely orientable codimension one foliations. Let be a Cr-foliation as above in a smooth manifold M, r ≧ 1, and let F0 be a closed leaf of . A neighborhood U of F0 is called a bicollar of F0 in this paper if there is a normal line bundle : U → F0 with respect to a fixed Riemannian metric on M such that each fibre of is transverse to .

Riemannian metrics and Laplacians for generalized smooth distributions

2019 ◽  
pp. 1-47
Author(s):  
Iakovos Androulidakis ◽  
Yuri Kordyukov
Keyword(s):  
Laplace Operator ◽  
Smooth Manifold ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Constant Rank ◽  
Singular Foliation ◽  
Pseudodifferential Calculus ◽  
Underlying Manifold

We show that any generalized smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Laplacian in the longitudinal pseudodifferential calculus of the smallest singular foliation which includes the distribution, we prove hypoellipticity.

scholarly journals Enlargeable length-structure and scalar curvatures

2021 ◽  
Author(s):  
Jialong Deng
Keyword(s):  
Scalar Curvature ◽  
Smooth Manifold ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Scalar Curvature ◽  
Smooth Manifolds ◽  
Connected Sum ◽  
Positive Scalar ◽  
Topological Manifolds ◽  
Arbitrary Dimensions

AbstractWe define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

On submersions of the complex indicatrix

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5081-5092
Author(s):  
Elena Popovicia
Keyword(s):  
Fixed Point ◽  
Complex Projective Space ◽  
Finsler Space ◽  
Hermitian Manifold ◽  
Almost Hermitian Manifold ◽  
Codimension One ◽  
Real Submanifold ◽  
Fundamental Equations ◽  
Complex Projective ◽  
Extrinsic Sphere

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

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