Contents......Page 14
Foreword......Page 8
Preface......Page 10
1.1.1 Pseudo-Riemannian metrics......Page 17
1.1.2.2 The Levi-Civita connection......Page 19
1.1.2.3 Curvature of a connection......Page 23
1.1.3 Calculus on a pseudo-Riemannian manifold......Page 24
1.2.1 The tangent and the normal spaces......Page 25
1.2.2 Intrinsic and extrinsic structures of a submanifold......Page 27
1.2.3 One-dimensional submanifolds: Curves......Page 30
1.2.3.2 Curvature of a curve......Page 31
1.2.3.3 Curves in surfaces and the Fr´enet equations......Page 32
1.2.4 Submanifolds of co-dimension one: Hypersurfaces......Page 33
1.3.1 Variation of a submanifold......Page 34
1.3.2 The first variation formula......Page 35
1.3.3 The second variation formula......Page 39
1.4 Exercises......Page 43
2.1 Intrinsic geometry of surfaces......Page 45
2.2 Graphs in Minkowski space......Page 48
2.3 The classification of ruled, minimal surfaces......Page 56
2.4 Weierstrass representation for minimal surfaces......Page 63
2.4.1 The definite case......Page 64
2.4.1.1 The case of dimension 3......Page 66
2.4.2 The indefinite case......Page 68
2.5 Exercises......Page 70
3.1 The pseudo-Riemannian space forms......Page 73
3.2.1 Equivariant hypersurfaces in pseudo-Euclidean space......Page 77
3.2.2 The minimal equation......Page 79
3.2.3 The definite case (ε, ε') = (1, 1)......Page 81
3.2.4 The indefinite positive case (ε, ε') = (−1, 1)......Page 82
3.2.5 The indefinite negative case (ε, ε') = (−1,−1)......Page 83
3.2.6 Conclusion......Page 84
3.3.1 Totally umbilic hypersurfaces in pseudo-space forms......Page 85
3.3.2 Equivariant hypersurfaces in pseudo-space forms......Page 89
3.3.3 Totally geodesic and isoparametric solutions......Page 91
3.3.4 The spherical case (ε, ε', ε") = (1, 1, 1)......Page 92
3.3.5 The ”elliptic hyperbolic” case (ε, ε', ε") = (1,−1,−1)......Page 94
3.3.6 The ”hyperbolic hyperbolic” case (ε, ε', ε") = (−1,−1, 1)......Page 96
3.3.7.1 The positive case......Page 97
3.3.8.1 The region {|z1| > |z2|}......Page 98
3.3.8.2 The region {|z1| < |z2|}......Page 99
3.3.9 Conclusion......Page 100
3.4 Exercises......Page 102
4.1 The complex pseudo-Euclidean space......Page 105
4.2 The general definition......Page 107
4.3 Complex space forms......Page 111
4.3.1 The case of dimension n = 1......Page 115
4.4.1 The canonical symplectic structure of the cotangent bundle T∗M......Page 116
4.4.2 An almost complex structure on the tangent bundle TM of a manifold equipped with an affine connection......Page 118
4.4.3 Identifying T∗M and TM and the Sasaki metric......Page 120
4.4.4 A complex structure on the tangent bundle of a pseudo-K¨ahler manifold......Page 122
4.4.5 Examples......Page 124
4.5 Exercises......Page 125
5.1 Complex submanifolds......Page 127
5.2 Lagrangian submanifolds......Page 129
5.3 Minimal Lagrangian surfaces in C2 with neutral metric......Page 130
5.4 Minimal Lagrangian submanifolds in Cn......Page 132
5.4.1 Lagrangian graphs......Page 134
5.4.2 Equivariant Lagrangian submanifolds......Page 136
5.4.3 Lagrangian submanifolds from evolving quadrics......Page 139
5.5 Minimal Lagrangian submanifols in complex space forms......Page 143
5.5.1 Lagrangian and Legendrian submanifolds......Page 144
5.5.2 Equivariant Legendrian submanifolds in odd-dimensional space forms......Page 149
5.5.3.1 The minimal equation......Page 153
5.5.3.3 The spherical case ε = 1......Page 155
5.5.3.4 The hyperbolic case ε = −1......Page 158
5.6 Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface......Page 159
5.6.1 Rank one Lagrangian surfaces......Page 160
5.6.2 Rank two Lagrangian surfaces......Page 162
5.7 Exercises......Page 164
6.1.1 Hypersurfaces in pseudo-Euclidean space......Page 167
6.1.2 Complex submanifolds in pseudo-K¨ahler manifolds......Page 171
6.1.3 Minimal Lagrangian submanifolds in complex pseudo-Euclidean space......Page 172
6.2 Non-minimizing submanifolds......Page 174
Bibliography......Page 177
Index......Page 181