Rigidity of compact minimal submanifolds in a sphere
In this paper we study n-dimensional compact minimal submanifolds in S with scalar curvature S satisfying the pinching condition S > n(n − 2). We show that for p ≤ 2 these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension p ≥ 2, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. Theorem 3.1 and Corollary 4.1). We also show that the scalar curvature S of a non-totally geodesic n-dimensional non-negatively curved minimal submanifold in S with flat normal connection satisfies n(n − p − 1) ≤ S ≤ n(n − 2) (cf. Theorem 4.1). Since for a compact hypersurface M of S the normal connection is flat, we use the above estimate for a scalar curvature S of a non-negatively curved minimal hypersurface M in S to infer that either M is totally geodesic or else it is isometric
Minimal immersions of 3-dimensional sphere into spheres
Theorem 0.1 (do Carmo and Wallach, [7]). Let f; S"C->S? be an isometric minimal immersion. Then (i) there exists an integer d such that c=k(d). (ii) There exists a positive semi-definite matrix A of size (m(d)-\-\)x(m(d)-\-\) such that f is equivalent to Ao-\JrΛtd. (iίΐ) If n~2 or d^3, then A is the identity matrix. (iv) If n^ and d^>4, then A is parametrized by a compact convex body L in some finite dimensional vector space, dim L^ 18. If A is an interior point of L then N=m(d)y and if A is a boundary point of L then N
Reflection coefficient of inhomogeneous anisotropic impedance plane
Solved the problem of constructive synthesis of an inhomogeneous anisotropic impedance plane reflecting a homogeneous plane wave in two different preset directions on a matched and cross polarization. Obtained the reflection coefficients and expressions for the scattering diagrams. The impedance is implemented by the frequent array of orthogonal strips. The law of distribution of the reactance was obtained in an explicit form. It is shown that for arbitrarily given (unequal) directions of reflections on matched and cross polarizations, the orientation of the reactance strips turns out to be variable along the plane. Formulated restrictions on class of implemented scattering diagrams. Shown the results of numerical studies.