Saddle surface

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)

References

[a1]	J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455