Namespaces
Variants
Actions

Difference between revisions of "Saddle surface"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A generalization of a surface of negative curvature (cf. [[Negative curvature, surface of|Negative curvature, surface of]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830801.png" /> be a surface in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830802.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830803.png" /> defined by an immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830804.png" /> of a two-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830805.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830806.png" />. A plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830807.png" /> cuts off a crust from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830808.png" /> if among the components of the inverse image of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s0830809.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s08308010.png" /> there is one with a compact closure. The part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s08308011.png" /> that corresponds to this component is called a crust (see Fig.).
+
<!--
 +
s0830801.png
 +
$#A+1 = 12 n = 0
 +
$#C+1 = 12 : ~/encyclopedia/old_files/data/S083/S.0803080 Saddle surface
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A generalization of a surface of negative curvature (cf. [[Negative curvature, surface of|Negative curvature, surface of]]). Let $  M $
 +
be a surface in the $  3 $-
 +
dimensional Euclidean space $  E  ^ {3} $
 +
defined by an immersion $  f: W \rightarrow E  ^ {3} $
 +
of a two-dimensional manifold $  W $
 +
in $  E  ^ {3} $.  
 +
A plane $  \alpha $
 +
cuts off a crust from $  M $
 +
if among the components of the inverse image of the set $  M \setminus  \alpha $
 +
in $  W $
 +
there is one with a compact closure. The part of $  M $
 +
that corresponds to this component is called a crust (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083080a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083080a.gif" />
Line 5: Line 28:
 
Figure: s083080a
 
Figure: s083080a
  
The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083080/s08308012.png" /> is called a saddle surface if it is impossible to cut off a crust by any plane. Examples of saddle surfaces are a one-sheet hyperboloid, a hyperbolic paraboloid and a ruled surface. For a twice continuously-differentiable surface to be a saddle surface it is necessary and sufficient that at each point of the surface its [[Gaussian curvature|Gaussian curvature]] is non-positive. A surface for which all its points are saddle points (cf. [[Saddle point|Saddle point]]) is a saddle surface.
+
The surface $  M $
 +
is called a saddle surface if it is impossible to cut off a crust by any plane. Examples of saddle surfaces are a one-sheet hyperboloid, a hyperbolic paraboloid and a ruled surface. For a twice continuously-differentiable surface to be a saddle surface it is necessary and sufficient that at each point of the surface its [[Gaussian curvature|Gaussian curvature]] is non-positive. A surface for which all its points are saddle points (cf. [[Saddle point|Saddle point]]) is a saddle surface.
  
 
A saddle surface that is bounded by a rectifiable contour is, with respect to its intrinsic metric induced by the metric of the space, a two-dimensional manifold of non-positive curvature. A number of properties of surfaces of negative curvature can be generalized to the class of saddle surfaces, but it seems that these surfaces do not form such a natural class of surfaces as do convex surfaces.
 
A saddle surface that is bounded by a rectifiable contour is, with respect to its intrinsic metric induced by the metric of the space, a two-dimensional manifold of non-positive curvature. A number of properties of surfaces of negative curvature can be generalized to the class of saddle surfaces, but it seems that these surfaces do not form such a natural class of surfaces as do convex surfaces.
Line 11: Line 35:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.Ya. Bakel'man,  A.L. Verner,  B.E. Kantor,  "Introduction to differential geometry  "in the large" " , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.Z. Shefel',  "Studies on the geometry of saddle-like surfaces" , Novosibirsk  (1963)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.Ya. Bakel'man,  A.L. Verner,  B.E. Kantor,  "Introduction to differential geometry  "in the large" " , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.Z. Shefel',  "Studies on the geometry of saddle-like surfaces" , Novosibirsk  (1963)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C.C. Nitsche,  "Vorlesungen über Minimalflächen" , Springer  (1975)  pp. §455</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C.C. Nitsche,  "Vorlesungen über Minimalflächen" , Springer  (1975)  pp. §455</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


A generalization of a surface of negative curvature (cf. Negative curvature, surface of). Let $ M $ be a surface in the $ 3 $- dimensional Euclidean space $ E ^ {3} $ defined by an immersion $ f: W \rightarrow E ^ {3} $ of a two-dimensional manifold $ W $ in $ E ^ {3} $. A plane $ \alpha $ cuts off a crust from $ M $ if among the components of the inverse image of the set $ M \setminus \alpha $ in $ W $ there is one with a compact closure. The part of $ M $ that corresponds to this component is called a crust (see Fig.).

Figure: s083080a

The surface $ M $ is called a saddle surface if it is impossible to cut off a crust by any plane. Examples of saddle surfaces are a one-sheet hyperboloid, a hyperbolic paraboloid and a ruled surface. For a twice continuously-differentiable surface to be a saddle surface it is necessary and sufficient that at each point of the surface its Gaussian curvature is non-positive. A surface for which all its points are saddle points (cf. Saddle point) is a saddle surface.

A saddle surface that is bounded by a rectifiable contour is, with respect to its intrinsic metric induced by the metric of the space, a two-dimensional manifold of non-positive curvature. A number of properties of surfaces of negative curvature can be generalized to the class of saddle surfaces, but it seems that these surfaces do not form such a natural class of surfaces as do convex surfaces.

References

[1] I.Ya. Bakel'man, A.L. Verner, B.E. Kantor, "Introduction to differential geometry "in the large" " , Moscow (1973) (In Russian)
[2] S.Z. Shefel', "Studies on the geometry of saddle-like surfaces" , Novosibirsk (1963) (In Russian)

Comments

References

[a1] J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455
How to Cite This Entry:
Saddle surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saddle_surface&oldid=48606
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article