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.
| Published in |
Pure and Applied Mathematics Journal (Volume 6, Issue 3-1)
This article belongs to the Special Issue Advanced Mathematics and Geometry |
| DOI | 10.11648/j.pamj.s.2017060301.12 |
| 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 |
Curvature, Differential Geometry, Geodesics, Manifolds, Parametrized, Surface
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| [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. |
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| [8] | T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, University of Georgia, 2016. |
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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
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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Download@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://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.},
year = {2017}
}
TY - JOUR T1 - An Introduction to Differential Geometry: The Theory of Surfaces AU - Kande Dickson Kinyua AU - Kuria Joseph Gikonyo Y1 - 2017/05/13 PY - 2017 N1 - https://doi.org/10.11648/j.pamj.s.2017060301.12 DO - 10.11648/j.pamj.s.2017060301.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 6 EP - 11 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.s.2017060301.12 AB - 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. VL - 6 IS - 3-1 ER -
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