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An Introduction to Differential Geometry: The Theory of Surfaces
Pure and Applied Mathematics Journal
Volume 6, Issue 3-1, June 2017, Pages: 6-11
Received: Feb. 6, 2017; Accepted: Feb. 14, 2017; Published: May 13, 2017
Authors
Kande Dickson Kinyua, Department of Mathematics, Moi University, Eldoret, Kenya; Department of Mathematics, Karatina University, Karatina, Kenya
Kuria Joseph Gikonyo, Department of Mathematics, Karatina University, Karatina, Kenya
Article Tools Abstract PDF (218KB)
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
Kande Dickson Kinyua, Kuria Joseph Gikonyo, An Introduction to Differential Geometry: The Theory of Surfaces, Pure and Applied Mathematics Journal. Special Issue: Advanced Mathematics and Geometry. Vol. 6, No. 3-1, 2017, pp. 6-11. doi: 10.11648/j.pamj.s.2017060301.12
References

K. D. Kinyua. An Introduction to Differentiable Manifolds, Mathematics Letters. Vol. 2, No. 5, 2016, pp. 32-35.

Lang, Serge, Introduction to Differentiable Manifolds, 2nd ed. Springer-Verlag New York. ISBN 0-387-95477-5, 2002.

M. Deserno, Notes on Difierential Geometry with special emphasis on surfaces in R3, Los Angeles, USA, 2004.

M. P. do-Carmo, Differential Geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Zealand, USA, 1976.

M. Raussen, Elementary Differential Geometry: Curves and Surfaces, Aalborg University, Denmark, 2008.

M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, Third Edition, Publish or Perish Inc., Houston, USA, 1999.

R. Palais, A Modern Course on Curves and Surfaces, 2003.

T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, University of Georgia, 2016.

V. G. Ivancevic and T. T. Ivancevic Applied Differential Geometry: A Modern Introduction, World Scientific Publishing Co. Pte. Ltd., Toh Tuck Link, Singapore, 2007.

W. Zhang, Geometry of Curves and Surfaces, Mathematics Institute, University of Warwick, 2014.
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