View source for Saddle surface

<!--
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" />

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|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.

====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>

====Comments====

====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>