scholarly journals Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1802
Author(s):  
Alex Iosevich ◽  
Krystal Taylor ◽  
Ignacio Uriarte-Tuero
Keyword(s):  
Fixed Point ◽  
Riemannian Manifold ◽  
Hausdorff Dimension ◽  
Riemannian Manifolds ◽  
Riemannian Metric ◽  
Compact Set ◽  
Geometric Configurations ◽  
Wide Range ◽  
D 12 ◽  
Euclidean Setting

Let M be a compact d-dimensional Riemannian manifold without a boundary. Given a compact set E⊂M, we study the set of distances from the set E to a fixed point x∈E. This set is Δρx(E)={ρ(x,y):y∈E}, where ρ is the Riemannian metric on M. We prove that if the Hausdorff dimension of E is greater than d+12, then there exist many x∈E such that the Lebesgue measure of Δρx(E) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. We give a simple proof of the Peres–Schlag result and generalize it to a wide range of distance type functions. Moreover, we extend our result to the setting of chains studied in our previous work and obtain a pinned estimate in this context.

Poisson–Voronoi Tessellation on a Riemannian Manifold

2020 ◽  
Author(s):  
Pierre Calka ◽  
Aurélie Chapron ◽  
Nathanaël Enriquez
Keyword(s):  
Fixed Point ◽  
Asymptotic Expansion ◽  
Riemannian Manifold ◽  
Poisson Point Process ◽  
Voronoi Tessellation ◽  
Voronoi Cell ◽  
Higher Dimension ◽  
Change Of Variables ◽  
The Mean ◽  
Euclidean Setting

Abstract In this paper, we consider a Riemannian manifold $M$ and the Poisson–Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda $ on $M$. We obtain a two-term asymptotic expansion, when $\lambda $ goes to infinity, of the mean number of vertices of the Voronoi cell associated with $x_0$. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of $M$ at $x_0$. This settles with the proper and rigorous frame the former 2D statement from [ 19] and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.

scholarly journals The Degenerate Special Lagrangian Equation on Riemannian Manifolds

2018 ◽  
Vol 2020 (9) ◽  
pp. 2832-2863
Author(s):  
Matthew Dellatorre
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Riemannian Manifolds ◽  
Duality Theory ◽  
Continuous Solutions ◽  
Lagrangian Equation ◽  
Base Manifold ◽  
Special Lagrangian ◽  
Euclidean Setting ◽  
Special Lagrangian Equation

Abstract We show that the degenerate special Lagrangian equation (DSL), recently introduced by Rubinstein–Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi–Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey–Lawson, in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence continuous geodesics in the space of positive Lagrangians.

scholarly journals The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

2015 ◽  
Vol 69 (1) ◽  
pp. 91
Author(s):  
Mariusz Plaszczyk
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Present Note ◽  
Riemannian Metric ◽  
Tensor Fields ◽  
Cotangent Bundles ◽  
Natural Transformations ◽  
The One

If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one J<sup>r</sup>TM → J<sup>r</sup>T*M between the r-th order prolongation J<sup>r</sup>TM of tangent TM and the r-th order prolongation J<sup>r</sup>T*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps D<sub>M</sub>(g) : J<sup>r</sup>TM → J<sup>r</sup>T*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.

scholarly journals 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

2020 ◽  
pp. 117-120
Author(s):  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Topological Structure ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Sectional Curvature ◽  
Levi Civita Connection ◽  
Connected Lie Group

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. 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].

Golden Riemannian structures on the tangent bundle with g-natural metrics

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2543-2554
Author(s):  
E. Peyghan ◽  
F. Firuzi ◽  
U.C. De
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Riemannian Metric

Starting from the g-natural Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,g), we construct a family of the Golden Riemannian structures ? on the tangent bundle (TM,G). Then we investigate the integrability of such Golden Riemannian structures on the tangent bundle TM and show that there is a direct correlation between the locally decomposable property of (TM,?,G) and the locally flatness of manifold (M,g).

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

scholarly journals NEI Modelling of the ISM - Turbulent Dissipation and Hausdorff Dimension

2009 ◽  
Vol 5 (H15) ◽  
pp. 468-469 ◽  
Author(s):  
Miguel A. de Avillez ◽  
Dieter Breitschwerdt
Keyword(s):  
High Resolution ◽  
Hausdorff Dimension ◽  
Dissipative Structures ◽  
Self Similarity ◽  
Extended Self ◽  
Turbulent Dissipation ◽  
Independent Information ◽  
Ionization Structure ◽  
Wide Range ◽  
Self Gravity

AbstractHigh-resolution non-ideal magnetohydrodynamical simulations of the turbulent magnetized ISM, powered by supernovae types Ia and II at Galactic rate, including self-gravity and non-equilibriuim ionization (NEI), taking into account the time evolution of the ionization structure of H, He, C, N, O, Ne, Mg, Si, S and Fe, were carried out. These runs cover a wide range (from kpc to sub-parsec) of scales, providing resolution independent information on the injection scale, extended self-similarity and the fractal dmension of the most dissipative structures.

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

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