Remarks on Manifolds with Two-Sided Curvature Bounds

Vitali Kapovitch; Alexander Lytchak

doi:10.1515/agms-2020-0122

Remarks on Manifolds with Two-Sided Curvature Bounds

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0122 ◽

2021 ◽

Vol 9 (1) ◽

pp. 53-64

Author(s):

Vitali Kapovitch ◽

Alexander Lytchak

Keyword(s):

Distance Functions ◽

Cut Locus ◽

Bounded Curvature ◽

Curvature Bounds ◽

Positive Reach

Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

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Computing curvature bounds for bounded curvature subdivision

Computer Aided Geometric Design ◽

10.1016/s0167-8396(01)00041-3 ◽

2001 ◽

Vol 18 (5) ◽

pp. 455-461 ◽

Cited By ~ 8

Author(s):

Jörg Peters ◽

Georg Umlauf

Keyword(s):

Bounded Curvature ◽

Curvature Bounds

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Analytic stratifications and the cut locus of a class of distance functions

Israel Journal of Mathematics ◽

10.1007/bf02773600 ◽

2006 ◽

Vol 154 (1) ◽

pp. 61-91 ◽

Cited By ~ 2

Author(s):

Paolo Albano ◽

Antonio Bove

Keyword(s):

Distance Functions ◽

Cut Locus

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Assessing flood risk and detecting changes of salt water inflow in a coastal micro-tidal brackish marsh using GIS

Oceanological and Hydrobiological Studies ◽

10.2478/v10009-008-0011-3 ◽

2008 ◽

Vol 37 (3) ◽

Author(s):

Jacek Urbański ◽

Agata Ślimak

Keyword(s):

Sea Level Rise ◽

Sea Level ◽

Flood Risk ◽

Salt Water ◽

Brackish Marsh ◽

Distance Functions ◽

Water Inflow ◽

Small Range ◽

Natural Environments ◽

Baltic Coast

Assessing flood risk and detecting changes of salt water inflow in a coastal micro-tidal brackish marsh using GISIn order to assess changes in salt water inflow and potential flood risks due to sea level rise in a micro-tidal Beka brackish marsh on the Polish Baltic Coast GIS was used. Such wetlands are important elements of coastal zone natural environments. Creating a geodatabase within a GIS system makes it possible to carry out broad analyses of complex systems, such as coastal wetlands. The results indicate that a 40 cm sea-level rise would considerably increase the frequency of flooding in the investigated area, in part because of the small range of the annual sea level oscillations there. A map of the index of changes in saltwater inflow, created with the help of cost-weighted distance (functions), shows that changes which have occurred along the shore, consisting of filling in the drainage channel outlets, have likely had a significant impact on the vegetation of the area.

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Some Considerations on the Cut Locus of a Riemannian Manifold

Geometry of Geodesics and Related Topics ◽

10.2969/aspm/00310029 ◽

2018 ◽

Cited By ~ 1

Author(s):

Jin-ichi Itoh

Keyword(s):

Riemannian Manifold ◽

Cut Locus

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Using clusterization principles for the selection of repair sections of main oil pipelines based on diagnostic data

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2011.1.019 ◽

2011 ◽

Vol 8 (1) ◽

pp. 201-210

Author(s):

R.M. Bogdanov

Keyword(s):

Cluster Analysis ◽

Defect Density ◽

Density Variation ◽

Distance Functions ◽

Oil Pipeline ◽

Oil Pipelines ◽

Classification Of Images ◽

Main Oil Pipeline ◽

Selection Of

The problem of determining the repair sections of the main oil pipeline is solved, basing on the classification of images using distance functions and the clustering principle, The criteria characterizing the cluster are determined by certain given values, based on a comparison with which the defect is assigned to a given cluster, procedures for the redistribution of defects in cluster zones are provided, and the cluster zones parameters are being changed. Calculations are demonstrating the range of defect density variation depending on pipeline sections and the universal capabilities of linear objects configuration with arbitrary density, provided by cluster analysis.

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Some fixed point results for (β-ψ1-ψ2)-contractive conditions in ordered b-metric-like spaces

Filomat ◽

10.2298/fil1714587p ◽

2017 ◽

Vol 31 (14) ◽

pp. 4587-4612 ◽

Cited By ~ 2

Author(s):

S.K. Padhan ◽

Rao Jagannadha ◽

Hemant Nashine ◽

R.P. Agarwal

Keyword(s):

Fixed Point ◽

Boundary Value Problems ◽

Fourth Order ◽

Metric Spaces ◽

Contractive Mapping ◽

Existence Of Solution ◽

Distance Functions ◽

Point Boundary ◽

Contractive Mappings ◽

Altering Distance

This paper extends and generalizes results of Mukheimer [(?,?,?)-contractive mappings in ordered partial b-metric spaces, J. Nonlinear Sci. Appl. 7(2014), 168-179]. A new concept of (?-?1-?2)-contractive mapping using two altering distance functions in ordered b-metric-like space is introduced and basic fixed point results have been studied. Useful examples are illustrated to justify the applicability and effectiveness of the results presented herein. As an application, the existence of solution of fourth-order two-point boundary value problems is discussed and rationalized by a numerical example.

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