A long standing conjecture is that the Besicovitch triangle, i.e., an equilateral triangle with side √ 28/27, is a worm-cover. We will show that indeed there exists a class of isosceles triangles, that includes the above equilateral triangle, where each triangle from the class is a worm-cover. These triangles are defined so that the shortest symmetric z-arc stretched from side to side and touching the base would have length one.
Let h be the second fundamental form of a compact submanifold M of the quaternion projective space HP n (Y). For any unit vector u^TM, set δ(u) = \\h(u,u)\\ 2 . We determine all compact totally complex submanifolds of HP 71 (I) (resp. all compact totally real minimal submanifolds of HP n (l)) satisfying condition δ(u)^-j-(resp.
δ(u)^-)for all unit vectors 1. Introduction.Let M be a smooth m-dimensional Riemannian manifold isometrically immersed in an (m+jfr)-dimensional Riemannian manifold M. Let h denote the second fundamental form of this immersion. For each IGM, /ι is a bilinear mapping from TM x xTM x into TMi, where TM X is the tangent space of M at x and TMi is the normal space. We denote by S(x) the square of the length of h at IGM. By Gauss' equation we have S(x)=m(m-1)-p(x), whenever M is immersed as a minimal submanifold of S m+P (l) with scalar curvature p(x) at x in M. Therefore S(x) is an intrinsic invariant of M.In 1968, J. Simons [12] discovered for the class of compact minimal m-dimensional submanifolds of the unit (m+£)-sphere that the totally geodesic submanifolds are isolated in the following sense: If S(x)
In [1], the first and third authors investigated the motion of barbells on surfaces of constant curvature. It is natural to extend this study to pendulums.We define the notion of a pendulum on a surface of constant curvature and study the motion of a mass at a fixed distance from a pivot. We consider some special cases for the pendulum.Case 1: a pivot that moves with constant speed along a fixed geodesic. Case 2: a pivot that undergoes acceleration along a fixed geodesic.
We consider totally complex submanifolds of the Cayley projective plane with estimates on the length squared of the second fundamental form. We determine those bounds for which the second fundamental form is parallel and for which the submanifold is totally geodesic. The case of totally real submanifolds is also included. O. INTRODUCTION Let M be a smooth m-dimensional Riemannian manifold isometrically immersed in an (m + p)-dimensional Riemannian manifold N. Let h denote the second fundamental form of this immersion. For each x ~ M, h is a bilinear mapping from TxM x TxM into TxM l, where T~M is the tangent space of M, and TxM ± is the normal space. We denote by S(x) the length squared of h at x E M. The Gauss equation shows that S(x) = m(m -1) -fl(x), where fl(x) is the scalar curvature of M at x. In this way S(x) is an intrinsic invariant of M. In [13] Simons showed that if M is minimal in S re+p, the totally geodesic submanifolds are isolated in so far that if S(x) ~< m/(2 -1/p) for all x ~ M, then S(x) = 0 identically on M, and M is totally geodesic. All minimal submanifolds of S "+p satisfying S(x) = m/(2 -l/p) were determined in [7]. Subsequently, analogous results for various minimal submanifolds of the complex and quaternionic projective space were obtained.Consider now the unit tangent bundle UM. We set 6(u) = IIh(u, u)l12 for u~ UM. Unlike S(x), 6(u) is not an intrinsic invariant of M, but can be considered as a natural measure of the degree to which an immersion fails to be totally geodesic.For CP n, Ros [11] showed that ifM is a compact Kaehler submanifold and if 6(u) < ¼, for any u ~ UM, then M is totally geodesic. He also lists those Kaehler submanifolds satisfying the condition maxu~vu{f(u)} = ¼. Coulton and Gauchman proved the same estimate for totally complex submanifolds of HP n, the quaternionic projective space [8]. Recall that totally complex submanifolds of HP ~ inherit a KaeMer structure in a natural way. Tsukada [14] has given a classification of parallel submanifolds of HP" which satisfy 6(u) = ¼. Analogous results have also been obtained [8] in the case of totally real immersions into HP n.There remains the rank 1 compact irreducible symmetric space, the Cayley Geometriae Dedicata 33: [265][266][267][268][269][270][271][272][273][274][275] 1990.
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