Estimated Upper Bound Of The Convex Sets Often Surfaces Weeks Genus

This dissertation investigates firstly the sufficient and necessary conditions for Lie groups to be of unimodula, and for homogeneous spaces to have invariant measure. We give new proofs of these properties. Then we prove that the group I(Xcn) of isometries of space form Xcn is unimodular and indeed the principle bundle O(Xcn) with structure group O(n; R). It also gives an explanation of the moving frame methods in Xcn. Furthermore, we derive the kinematic density of a homogeneous space of a class of homogeneous hypersurface in Xcn with two constant principal curvatures.Let K be a domain in the constant curvature surface Xc2, A and L be the area of K and the length of (?)K, respectively. Then the isoperimetric inequality of K is L2-4πA+cA2≥0. The equality sign holds if and only if K is a geodesic disc. The quantityΔ(K)=L2-4πA+cA2 is called the isoperimetric deficit of K. The lower bound of the isoperimetric deficit have been obtained by D. Klain, J. Zhou and F. Chen. We know little about the upper bound of the isoperimetric deficit. In the plane case,O. Bottema given an upper bound of the isoperimetric deficit for an oval domain. We generalize the result to the case Xc2.Theorem. Let K be a convex domain in Xc2 with smooth and strictly convex boundary (?)K. When c<0, we assume that the geodesic curvature of (?)K satisfyρ(x) denotes the curvature radius at x∈(?)K. LetρM andρm be the maximum and minimum of p respectively, then we have The equality sign holds if and only if K is a geodesic disc. We also haveAt the end, we give a new proof of the Blaschke's rolling theorem in Xc2. The theorem is necessary for the estimating of the upper bound of the isoperimetric deficit.