Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

On the total mean curvature of a nonrigid surface

Using the Green’s theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field, which immediately yields the following well-known theorem: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.     

A generalization of Bonnet’s theorem on Darboux surfaces

The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to n-dimensional hypersurfaces of Euclidean spaces.     

Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C~2-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C~2-smooth curves on surfaces. We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane.     

Euclidean Complete Affine Surfaces with Constant Affine Mean Curvature

The purpose of this paper is to prove that alocally strongly convex, Euclidean complete surface with constantaffine mean curvature is also affine complete. Consequently weobtain a classification of locally strongly convex, Euclideancomplete surfaces with constant affine mean curvature.     

A bound for the principal radii of curvature of a closed convex surface with prescribed function of the relative radii of curvature

We prove the existence of an upper bound for the radii of normal curvature of a ciosed convex surface in three-dimensional Euclidean space with a prescribed function of the relative radii of curvature.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 82–92.     

Capillary surfaces of constant mean curvature in a right solid cylinder

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the solid cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to conclude that the surface is part of a plane or a catenoid.     

Saddle surfaces in singular spaces

The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel' on compactness of saddle surfaces in a Euclidean space.

     

Differential invariants of surfaces

The algebra of differential invariants of a suitably generic surface S⊂R³, under either the usual Euclidean or equi-affine group actions, is shown to be generated, through invariant differentiation, by a single differential invariant. For Euclidean surfaces, the generating invariant is the mean curvature, and, as a consequence, the Gauss curvature can be expressed as an explicit rational function of the invariant derivatives, with respect to the Frenet frame, of the mean curvature. For equi-affine surfaces, the generating invariant is the third order Pick invariant. The proofs are based on the new, equivariant approach to the method of moving frames.     

A Condition for a Closed One-Form to Be Exact

A condition for a closed one-form to be exact, the one-form having values in Euclidean space, on a compact surface without boundary, is given in the case where the surface has suitable differentiable automorphisms. Tori and hyperelliptic curves, with holomorphic automorphisms, are in this case. A local representation formula for surfaces in Euclidean space is then globalized. A condition for a local surface of constant mean curvature to be global, can be written using a harmonic Gauss map.     

常曲率平面上的逆Bonnesen型不等式

利用积分几何中估计包含测度的思想给出常曲率平面上一些新的逆Bonnesen型不等式.这些不等式在欧氏平面上为著名的Bottema不等式的改进形式与新的逆Bonnesen型不等式.     

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.     

The classification of homotopy classes of bounded curvature paths

A bounded curvature path is a continuously differentiable piecewise C² path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our work in [3], [2] and [4] this work finishes a program started by Lester Dubins in [6] in 1961.     

Some sufficient conditions for immersion in the large of metrics of positive curvature

It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallL _p (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavior of the geodesic boundary curvature).Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 51–59, 1991.     

Smooth approximation of polyhedral surfaces regarding curvatures

In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.The present paper is a detailed version of the short announcement [3].     

Positivity of curvature and convexity of faces

Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons.     

Generalization of theorems of Darboux-Sauer and Pogorelov

The Darboux-Sauer theorem establishes a relationship between infinitely small deformations (i.s.d.) of an arbitrary surface in Euclidean space and the i.s.d. of its image under a projective transformation of the space. Pogorelov established a similar relationship between the i.s.d. of a surface in a space of constant curvature and the i.s.d. of its image under a projective (geodesic) mapping of the space into a Euclidean space. In the present paper we show that both of these theorems are easily obtained from two simple results concerning Killing vector fields (i.s. motions of a space). The first of these results consists in the proportionality of the covariant components of such fields in spaces in geodesic correspondence (in the coordinates carried over by this correspondence). The second theorem establishes a relationship between i.s.d. and Killing fields in spaces of constant curvature, thereby generalizing the well-known construction of the field of rotations for i.s.d. in a Euclidean space. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 63–67, 1974.     

An extension of a theorem of Serrin to graphs in warped products

In this article we extend a well known theorem of J. Serrin about existence and uniqueness of graphs of constant mean curvature in Euclidean space to a broad class of Riemannian manifolds. Our result also generalizes several others proved recently and includes the new case of Euclidean “rotational” graphs with constant mean curvature.     

Algebraic surfaces defined by a homogeneous polynomial

We study algebraic surfaces in three-dimensional Euclidean space given explicitly by a homogeneous polynomial. We ascertain necessary and sufficient conditions for the Gaussian curvature to be of constant sign. We give an estimate on the absolute curvature of the surface in terms of the number of distinct linear factors occurring in the decomposition of the polynomial.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 3–10.     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.