Minimal submanifolds in pseudo-Riemannian geometry / Henri Anciaux.

• Machine generated contents note: 1.Submanifolds in pseudo-Riemannian geometry -- 1.1.Pseudo-Riemannian manifolds -- 1.1.1.Pseudo-Riemannian metrics -- 1.1.2.Structures induced by the metric -- 1.1.3.Calculus on a pseudo-Riemannian manifold -- 1.2.Submanifolds -- 1.2.1.The tangent and the normal spaces -- 1.2.2.Intrinsic and extrinsic structures of a submanifold -- 1.2.3.One-dimensional submanifolds: Curves -- 1.2.4.Submanifolds of co-dimension one: Hypersurfaces -- 1.3.The variation formulae for the volume -- 1.3.1.Variation of a submanifold -- 1.3.2.The first variation formula -- 1.3.3.The second variation formula -- 1.4.Exercises -- 2.Minimal surfaces in pseudo-Euclidean space -- 2.1.Intrinsic geometry of surfaces -- 2.2.Graphs in Minkowski space -- 2.3.The classification of ruled, minimal surfaces -- 2.4.Weierstrass representation for minimal surfaces -- 2.4.1.The definite case -- 2.4.2.The indefinite case -- 2.4.3.A remark on the regularity of minimal surfaces -- 2.5.Exercises -- 3.Equivariant minimal hypersurfaces in space forms -- 3.1.The pseudo-Riemannian space forms -- 3.2.Equivariant minimal hypersurfaces in pseudo-Euclidean space -- 3.2.1.Equivariant hypersurfaces in pseudo-Euclidean space -- 3.2.2.The minimal equation -- 3.2.3.The definite case (ε,ε') = (1,1) -- 3.2.4.The indefinite positive case (ε,ε') = (-1,1) -- 3.2.5.The indefinite negative case (ε,ε') = (-1,-1) -- 3.2.6.Conclusion -- 3.3.Equivariant minimal hypersurfaces in pseudo-space forms -- 3.3.1.Totally umbilic hypersurfaces in pseudo-space forms -- 3.3.2.Equivariant hypersurfaces in pseudo-space forms -- 3.3.3.Totally geodesic and isoparametric solutions -- 3.3.4.The spherical case (ε,ε',ε") = (1,1,1) -- 3.3.5.The "elliptic hyperbolic" case (ε,ε',ε") = (1,-1,-1) -- 3.3.6.The "hyperbolic hyperbolic" case (ε,ε',ε") = (-1,-1,1) -- 3.3.7.The "elliptic" de Sitter case (ε,ε',ε") = (-1,1,1) -- 3.3.8.The "hyperbolic" de Sitter case (ε,ε',ε") = (1,-1,1) -- 3.3.9.Conclusion -- 3.4.Exercises -- 4.Pseudo-Kahler manifolds -- 4.1.The complex pseudo-Euclidean space -- 4.2.The general definition -- 4.3.Complex space forms -- 4.3.1.The case of dimension n = 1 -- 4.4.The tangent bundle of a psendo-Kahler manifold -- 4.4.1.The canonical symplectic structure of the cotangent bundle TM -- 4.4.2.An almost complex structure on the tangent bundle TM of a manifold equipped with an affine connection -- 4.4.3.Identifying TM and TM and the Sasaki metric -- 4.4.4.A complex structure on the tangent bundle of a pseudo-Kahler manifold -- 4.4.5.Examples -- 4.5.Exercises -- 5.Complex and Lagrangian submanifolds in pseudo-Kahler manifolds -- 5.1.Complex submanifolds -- 5.2.Lagrangian submanifolds -- 5.3.Minimal Lagrangian surfaces in C2 with neutral metric -- 5.4.Minimal Lagrangian submanifolds in Cn -- 5.4.1.Lagrangian graphs -- 5.4.2.Equivariant Lagrangian submanifolds -- 5.4.3.Lagrangian submanifolds from evolving quadrics -- 5.5.Minimal Lagrangian submanifols in complex space forms -- 5.5.1.Lagrangian and Legendrian submanifolds -- 5.5.2.Equivariant Legendrian submanifolds in odd-dimensional space forms -- 5.5.3.Minimal equivariant Lagrangian submanifolds in complex space forms -- 5.6.Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface -- 5.6.1.Rank one Lagrangian surfaces -- 5.6.2.Rank two Lagrangian surfaces -- 5.7.Exercises -- 6.Minimizing properties of minimal submanifolds -- 6.1.Minimizing submanifolds and calibrations -- 6.1.1.Hypersurfaces in pseudo-Euclidean space -- 6.1.2.Complex submanifolds in pseudo-Kahler manifolds -- 6.1.3.Minimal Lagrangian submanifolds in complex pseudo-Euclidean space -- 6.2.Non-minimizing submanifolds.

Ämnesord

Riemannian manifolds.  (LCSH)

Minimal submanifolds.  (LCSH)

Klassifikation

QA649 (LCC)

516.373 (DDC)

Tel (kssb/8 (machine generated))