### An Introduction to Differential Geometry: The Theory of Surfaces

Received: 6 February 2017     Accepted: 14 February 2017     Published: 13 May 2017
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.

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.
• 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

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

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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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://article.sciencepublishinggroup.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.},
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}
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Author Information
• Department of Mathematics, Moi University, Eldoret, Kenya

• Department of Mathematics, Karatina University, Karatina, Kenya

• Sections