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THE METRIC DIMENSION OF METRIC MANIFOLDS

Part of: Global differential geometry Spaces with richer structures Distance geometry

Published online by Cambridge University Press:  11 February 2015

MAJID HEYDARPOUR  and
SAEID MAGHSOUDI
MAJID HEYDARPOUR*
Affiliation:
Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran email [email protected]
SAEID MAGHSOUDI
Affiliation:
Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran email [email protected]
*
✉[email protected]
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Abstract

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In this paper we determine the metric dimension of $n$-dimensional metric $(X,G)$-manifolds. This category includes all Euclidean, hyperbolic and spherical manifolds as special cases.

Keywords

metric dimensionmetric (X, G)-manifoldhyperbolic and spherical spacesRiemannian manifoldgeometric space

MSC classification

Primary: 54E35: Metric spaces, metrizability
Secondary: 51K05: General theory 53C20: Global Riemannian geometry, including pinching
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015 , pp. 508 - 513
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Bau, S. and Beardon, A. F., ‘The metric dimension of metric spaces’, Comput. Methods Funct. Theory 13 (2013), 295–305.Google Scholar
Blumenthal, L. M., Theory and Applications of Distance Geometry (Clarendon Press, Oxford, 1953).Google Scholar
Boutin, D. L., ‘Determining sets, resolving sets, and the exchange property’, Graphs Combin. 25 (2009), 789–806.Google Scholar
Cceres, J., Hernando, C., Mora, M., Pelayo, I. M. and Puertas, M. L., ‘On the metric dimension of infinite graphs’, Electron. Notes Discrete Math. 35 (2009), 15–20.CrossRefGoogle Scholar
Chappell, G. C., Gimbel, J. and Hartman, C., ‘Bounds on the metric and partition dimension of a graph’, Ars Combin. 88 (2008), 349–366.Google Scholar
Harary, F. and Melter, R. A., ‘On the metric dimension of a graph’, Ars Combin. 2 (1976), 191–195.Google Scholar
Heydarpour, M. and Maghsoudi, S., ‘The metric dimension of geometic spaces’, Topology Appl. 178 (2014), 230–235.CrossRefGoogle Scholar
Khuller, S., Raghavachari, B. and Rosenfeld, A., ‘Landmarks in graphs’, Discrete Appl. Anal. 70 (1996), 217–229.CrossRefGoogle Scholar
Lee, J. M., Introduction to Smooth Manifolds, 2nd edn, Graduate Texts in Mathematics, 218 (Springer, New York, 2013).Google Scholar
Melter, R. A. and Tomescu, I., ‘Metric basis in digital geometry’, Comput. Vis. Graph. Image Process 25 (1984), 113–121.CrossRefGoogle Scholar
Ratcliffe, J. G., Foundations of Hyperbolic Manifolds (Springer, New York, 1994).CrossRefGoogle Scholar
Slater, P. J., ‘Leaves of trees’, Congr. Numer. 14 (1975), 549–559.Google Scholar