We obtain a pointwise inequality valid for all submanifolds of all real space forms with and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of , and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of in .
Introduction. For bounded domains in satisfying the cone condition there are many embedding and module structure theorem for Sobolev spaces which are of great importance in solving partial differential equations. Unfortunately, most of them are wrong on arbitrary unbounded domains or on open manifolds. On the other hand, just these theorems play a decisive role in foundations of nonlinear analysis on open manifolds and in solving partial differential equations. This was pointed out by the author...
∗Research supported in part by NSF grant INT-9903302.In previous work a hyperbolic twistor space over a paraquaternionic Kähler manifold was defined, the fibre being the hyperboloid model of the hyperbolic plane with constant curvature −1. Two almost complex structures were defined on this twistor space and their properties studied. In the present paper we consider a twistor space over a paraquaternionic Kähler manifold with fibre given by the hyperboloid of 1-sheet, the anti-de-Sitter plane...
In this note we study the Ledger conditions on the families of flag manifold , , constructed by N. R. Wallach in (Wallach, N. R., Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), 276–293.). In both cases, we conclude that every member of the both families of Riemannian flag manifolds is a D’Atri space if and only if it is naturally reductive. Therefore, we finish the study of made by D’Atri and Nickerson in (D’Atri, J. E., Nickerson, H. K., Geodesic...