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
Rigidity of Complete Minimal Submanifolds in Spheres
Let M be an n-dimensional complete minimal submanifold in an (n + p)-dimensional sphere 𝕊n+p, and let h be the second fundamental form of M. In this paper, it is shown that M is totally geodesic if the L2 norm of |h| on any geodesic ball of M is of less than quadratic growth and the Ln norm of |h| on M is less than a fixed constant. Further, under only the latter condition, we prove that M is totally geodesic. Moreover, we provide a sufficient condition for a complete stable minimal hypersurface to be totally geodesic.
Fascism and the right in Europe, 1919-1945
This new text places interwar European fascism squarely in its historical context and analyses its relationship with other right wing, authoritarian movements and regimes. Beginning with the ideological roots of fascism in pre-1914 Europe, Martin Blinkhorn turns to the problem-torn Europe of 1919 to 1939 in order to explain why fascism emerged and why, in some settings, it flourished while in others it did not. In doing so he considers not just the 'major' fascist movements and regimes of Italy and Germany but the entire range of fascist and authoritarian ideas, movements and regimes present in the Europe of 1919-1945.