Pure and Applied Mathematics Journal

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An Introduction to Differential Geometry: The Theory of Surfaces

Received: 06 February 2017    Accepted: 14 February 2017    Published: 13 May 2017
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Abstract

From a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an introduction to the theory of surfaces from differential geometry perspective.

DOI 10.11648/j.pamj.s.2017060301.12
Published in Pure and Applied Mathematics Journal (Volume 6, Issue 3-1, June 2017)

This article belongs to the Special Issue Advanced Mathematics and Geometry

Page(s) 6-11
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Curvature, Differential Geometry, Geodesics, Manifolds, Parametrized, Surface

References
[1] K. D. Kinyua. An Introduction to Differentiable Manifolds, Mathematics Letters. Vol. 2, No. 5, 2016, pp. 32-35.
[2] Lang, Serge, Introduction to Differentiable Manifolds, 2nd ed. Springer-Verlag New York. ISBN 0-387-95477-5, 2002.
[3] M. Deserno, Notes on Difierential Geometry with special emphasis on surfaces in R3, Los Angeles, USA, 2004.
[4] M. P. do-Carmo, Differential Geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Zealand, USA, 1976.
[5] M. Raussen, Elementary Differential Geometry: Curves and Surfaces, Aalborg University, Denmark, 2008.
[6] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Third Edition, Publish or Perish Inc., Houston, USA, 1999.
[7] R. Palais, A Modern Course on Curves and Surfaces, 2003.
[8] T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, University of Georgia, 2016.
[9] V. G. Ivancevic and T. T. Ivancevic Applied Differential Geometry: A Modern Introduction, World Scientific Publishing Co. Pte. Ltd., Toh Tuck Link, Singapore, 2007.
[10] W. Zhang, Geometry of Curves and Surfaces, Mathematics Institute, University of Warwick, 2014.
Author Information
  • Department of Mathematics, Moi University, Eldoret, Kenya; Department of Mathematics, Karatina University, Karatina, Kenya

  • Department of Mathematics, Karatina University, Karatina, Kenya

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  • APA Style

    Kande Dickson Kinyua, Kuria Joseph Gikonyo. (2017). An Introduction to Differential Geometry: The Theory of Surfaces. Pure and Applied Mathematics Journal, 6(3-1), 6-11. https://doi.org/10.11648/j.pamj.s.2017060301.12

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    ACS Style

    Kande Dickson Kinyua; Kuria Joseph Gikonyo. An Introduction to Differential Geometry: The Theory of Surfaces. Pure Appl. Math. J. 2017, 6(3-1), 6-11. doi: 10.11648/j.pamj.s.2017060301.12

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    AMA Style

    Kande Dickson Kinyua, Kuria Joseph Gikonyo. An Introduction to Differential Geometry: The Theory of Surfaces. Pure Appl Math J. 2017;6(3-1):6-11. doi: 10.11648/j.pamj.s.2017060301.12

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  • @article{10.11648/j.pamj.s.2017060301.12,
      author = {Kande Dickson Kinyua and Kuria Joseph Gikonyo},
      title = {An Introduction to Differential Geometry: The Theory of Surfaces},
      journal = {Pure and Applied Mathematics Journal},
      volume = {6},
      number = {3-1},
      pages = {6-11},
      doi = {10.11648/j.pamj.s.2017060301.12},
      url = {https://doi.org/10.11648/j.pamj.s.2017060301.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.s.2017060301.12},
      abstract = {From a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an introduction to the theory of surfaces from differential geometry perspective.},
     year = {2017}
    }
    

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