Minimal submanifolds in pseudo-Riemannian geometry

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 (& epsilon;, & epsilon;') = (1,1).
3.2.4. The indefinite positive case (& epsilon;, & epsilon;') = ( -1,1).
3.2.5. The indefinite negative case (& epsilon;, & epsilon;') = ( -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 (& epsilon;, & epsilon;', & epsilon;") = (1,1,1).
3.3.5. The "elliptic hyperbolic" case (& epsilon;, & epsilon;', & epsilon;") = (1,-1,-1).
3.3.6. The "hyperbolic hyperbolic" case (& epsilon;, & epsilon;', & epsilon;") = ( -1,-1,1).
3.3.7. The "elliptic" de Sitter case (& epsilon;, & epsilon;', & epsilon;") = ( -1,1,1).
3.3.8. The "hyperbolic" de Sitter case (& epsilon;, & epsilon;', & epsilon;") = (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.
Since the foundational work of Lagrange on the differential equation to be satisfied by a minimal surface of the Euclidean space, the theory of minimal submanifolds have undergone considerable developments, involving techniques from related areas, such as the analysis of partial differential equations and complex analysis. On the other hand, the relativity theory has led to the study of pseudo-Riemannian manifolds, which turns out to be the most general framework for the study of minimal submanifolds. However, most of the recent books on the subject still present the theory only in the Riemann