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assume automorphism bounded cohomology compact condition constant curvature convex coordinates Corollary corresponding curvature tensor curve defined Definition denote derivatives diffeomorphism differentiable space differential forms Differential Geometry differential operator dimension eigenvalue elliptic equivalent estimate Euclidean exists formula G-invariant given Hence Hermitian symmetric holomorphic homotopy hypersurface implies induced integrable invariant circle involutive isometric immersion isomorphism jet bundle Lemma Lie algebra linear maps Math metric minimal neighborhood nonnegative nonzero normal obtain orbit orthogonal orthonormal partial differential equations partitions of unity positive principal curvatures proof of Theorem Proposition prove pseudo-group real hypersurface reflective submanifolds resp respectively result Rham Ricci curvature Riemannian manifold satisfies second fundamental form sectional curvature sequence sheaf simply connected smooth function smooth partitions solution structure submanifolds subset subspace Suppose surface symmetric spaces tangent theory topology totally geodesic unique variables vector bundle vector field vector space zero