Distribution of minimal varieties in spheres in terms of the coordinate functions

Let M be a compact k-dimensional riemmanian manifold minimally immersed in the unit n-dimensional sphere S. It is easy to show that for any p ∈ S the boundary of the geodesic ball in S with radius π2 and center at p (in this case this boundary is an equator) must intercept the manifold M . When the codimension is 1, i.e. k = n − 1, it is known that the ricci curvature, is not greater than 1. We will prove that if the ricci curvature is not greater than 1− α 2 n−2 , then the boundary of every geodesic ball with radius cot−1(α) must intercept the manifold M . We give examples of manifols for which the radius cot−1(α) is optimal. Next, for any codimension, i.e. for any M ⊂ S, we find a number r1 that depends only on n such that for any collection of n + 1 points {pi} i=1 in S that constitutes an orthonormal basis of R, the union of the boundaries of the geodesic balls with radius r1 and center pi, i = 1, 2, . . . n+ 1, must intercept the manifold M . §