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Higher Dimensions and Codimensions
The Fundamental Equations for Submanifolds
E Further Results
7 other sections not shown
1-forms Addendum assume choose clearly closed curve Codazzi-Mainardi equations compact complete conformal map conjugate points conjugate value consider constant curvature constant mean curvature convex Corollary covariant critical point cut point define denote derivative diffeomorphism differential dxdy everywhere function f Gauss geodesic map given gives hence hypersurface imbedding implies inner product integral intersect isometry isoperimetric problem isothermal coordinate isothermal coordinate system Jacobi field Lemma M C N manifold of constant mean curvature minimal geodesic minimal surface minimum moving frame n-dimensional neighborhood non-zero normal bundle obtain oriented orthogonal orthonormal perpendicular piecewise plane principal curvatures Proposition prove result Riemannian manifold satisfies second fundamental form sectional curvatures shows solution sphere submanifold subspace Suppose tangent vectors tensor totally geodesic umbilics unit normal unit vector variation vector field vector field vector space zero