Skew loops and quadric surfaces

Abstract

Abstract.A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R 3 with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces. Mathematics Subject Classification (2000). Primary 53A04, 53A05; Secondary 53C45, 52A15.

Totally skew embeddings of manifolds

“…Totally skew submanifolds (TS-embeddings) are generalizations of skew loops and skew branes [13,15,16,33] which are submanifolds in an affine space without parallel tangent spaces (S-embeddings). Skew loops were first studied by B. Segre [29] and have connections to quadric surfaces (skew branes were studied by Lai [25]; see also [34]).…”

“…Skew loops were first studied by B. Segre [29] and have connections to quadric surfaces (skew branes were studied by Lai [25]; see also [34]). In particular, the first author and Solomon [15] showed that the absence of skew loops characterizes quadric surfaces of positive curvature (including ellipsoids), while the second author [33] ruled out the existence of skew branes on quadric hypersurfaces in any dimension; see also [36] and [30]. Note, however, that skew loops are only affinely invariant, whereas quadric surfaces are invariant under the more general class of projective transformations.…”

Existence and nonexistence of skew branes

“…Then g − c is a derivative, g(x) − c = −f ′ (x), and df + α = cdx. It follows that, in dimension one, Conjecture 3.1 holds as well: this the Cylinder Lemma of [6].…”

Central Cross-Sections Make Surfaces of Revolution Quadric

“…The generalization described in Remark 2.5 will remove the need to assume rotational symmetry. Still, the main theorem here yields a first negatively curved counterpart to the characterization of positively curved quadrics in [3].…”