Intro -- Contents -- Foreword -- Preface -- Chapter 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 -- 1.1.2.1 Volume -- 1.1.2.2 The Levi-Civita connection -- 1.1.2.3 Curvature of a connection -- 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 -- 1.2.3.1 Arc length -- 1.2.3.2 Curvature of a curve -- 1.2.3.3 Curves in surfaces and the Frénet equations -- 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 -- Chapter 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 -- 2.4.1.1 The case of dimension 3 -- 2.4.2 The indefinite case -- 2.4.3 A remark on the regularity of minimal surfaces -- 2.5 Exercises -- Chapter 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,&#x2212;1,&#x2212;1) -- 3.3.6 The "hyperbolic hyperbolic" case (ε, ε', ε") = (&#x2212;1,&#x2212;1, 1) -- 3.3.7 The "elliptic" de Sitter case (ε, ε', ε") = (&#x2212;1, 1, 1) -- 3.3.7.1 The positive case -- 3.3.8 The "hyperbolic" de Sitter case (ε, ε', ε") = (1,&#x2212;1, 1) -- 3.3.8.1 The region {|z1| > |z2|} -- 3.3.8.2 The region {|z1| < |z2|} -- 3.3.9 Conclusion -- 3.4 Exercises -- Chapter 4 Pseudo-Kähler 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 pseudo-Kähler manifold -- 4.4.1 The canonical symplectic structure of the cotangent bundle T&#x2217;M -- 4.4.2 An almost complex structure on the tangent bundle TM of a manifold equipped with an affine connection -- 4.4.3 Identifying T&#x2217;M and TM and the Sasaki metric -- 4.4.4 A complex structure on the tangent bundle of a pseudo-Kähler manifold -- 4.4.5 Examples -- 4.5 Exercises -- Chapter 5 Complex and Lagrangian submanifolds in pseudo-Kähler 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 -- 5.5.3.1 The minimal equation -- 5.5.3.2 Special solutions -- 5.5.3.3 The spherical case ε = 1 -- 5.5.3.4 The hyperbolic case ε = &#x2212;1 -- 5.5.3.5 Conclusion.

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 -- Chapter 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-Kähler manifolds -- 6.1.3 Minimal Lagrangian submanifolds in complex pseudo-Euclidean space -- 6.2 Non-minimizing submanifolds -- Bibliography -- Index.

Key Features:One of the first books exposing the theory of minimal submanifolds in the general setting of pseudo-Riemannian geometryAn accessible text suitable for students, yet of interest to researchers in differential geometry and mathematical physicsShowcases several of the most recent research results on the subject in a unified and easily understood presentation.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.