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), 2017. Published by Science Publishing Group |
Curvature, Differential Geometry, Geodesics, Manifolds, Parametrized, Surface
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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
@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 -