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Difference between revisions of "Normal section"

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''of a smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676201.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676202.png" /> in a direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676203.png" />''
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''of a smooth surface $\Phi$ at a point $P$ in a direction $\ell$''
  
The section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676204.png" /> by the plane passing through the normal to the surface (cf. [[Normal space (to a surface)|Normal space (to a surface)]]) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676205.png" /> and through the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676206.png" /> in the [[Tangent plane|tangent plane]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676207.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067620/n0676208.png" />. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see [[Curvature|Curvature]]; [[Normal curvature|Normal curvature]]). The method of studying the local structure by means of normal sections can be generalized to surfaces of arbitrary dimension and arbitrary codimension.
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The section of $\Phi$ by the plane passing through the normal to the surface (cf. [[Normal space (to a surface)]]) at $P$ and through the direction $\ell$ in the [[tangent plane]] to $\Phi$ at $P$. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see [[Curvature]]; [[Normal curvature]]). The method of studying the local structure by means of normal sections can be generalized to surfaces of arbitrary dimension and arbitrary codimension.
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "A course in differential geometry" , Springer  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.-Y. Chen,  "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "A course in differential geometry" , Springer  (1978)  (Translated from German)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  B.-Y. Chen,  "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR>
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</table>
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Revision as of 22:14, 15 December 2016

of a smooth surface $\Phi$ at a point $P$ in a direction $\ell$

The section of $\Phi$ by the plane passing through the normal to the surface (cf. Normal space (to a surface)) at $P$ and through the direction $\ell$ in the tangent plane to $\Phi$ at $P$. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see Curvature; Normal curvature). The method of studying the local structure by means of normal sections can be generalized to surfaces of arbitrary dimension and arbitrary codimension.


Comments

References

[a1] W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)
[a2] B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)
How to Cite This Entry:
Normal section. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_section&oldid=40020
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article