## Pseudo-Riemannian Geometry, [delta]-invariants and ApplicationsThe first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included.The second part of this book is on δ-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades. |

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### Contents

1 PseudoRiemannian Manifolds | 1 |

2 Basics on PseudoRiemannian Submanifolds | 25 |

3 Special PseudoRiemannian Submanifolds | 53 |

4 Warped Products and Twisted Products | 77 |

5 RobertsonWalker Spacetimes | 91 |

6 Hodge Theory Elliptic Differential Operators and Jacobis Elliptic Functions | 107 |

7 Submanifolds of Finite Type | 127 |

8 Total Mean Curvature | 161 |

13 δinvariants Inequalities and Ideal Immersions | 251 |

14 Some Applications of δinvariants | 279 |

15 Applications to Kahler and ParaKahler geometry | 305 |

16 Applications to Contact Geometry | 335 |

17 Applications to Affine Geometry | 345 |

18 Applications to Riemannian Submersions | 377 |

19 Nearly Kahler Manifolds and Nearly Kahler S61 | 393 |

20 δ2ideal Immersions | 417 |

9 PseudoKahler Manifolds | 183 |

10 ParaKahler Manifolds | 205 |

11 PseudoRiemannian Submersions | 227 |

12 Contact Metric Manifolds and Submanifolds | 241 |

Bibliography | 439 |

463 | |

473 | |

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### Common terms and phrases

afﬁne biharmonic called centroafﬁne Chen compact Riemannian manifold complex space form constant curvature Corollary cosh curvature tensor curve deﬁned Deﬁnition denote differential eigenvalue Einstein embedding equality sign equation Euclidean space ﬁat ﬁnd ﬁnite type ﬁrst Gauss Geom geometry graph hypersurface Hence holds identically horizontal hypersphere hypersurface ideal immersion implies indeﬁnite inequality invariant isometric immersion Kahler manifold Lagrangian submanifold Lemma Levi-Civita connection Math mean curvature vector minimal immersion minimal submanifold n-dimensional n-manifold N1 f N2 nonzero obtain open portion orthogonal orthonormal basis orthonormal frame para-Kahler parallel mean curvature point p G N principal curvatures Proof Proposition pseudo pseudo-Euclidean pseudo-Riemannian manifold pseudo-Riemannian submanifold real number real space form Ricci Riemannian manifold Riemannian submersion Sasakian space forms satisﬁes the equality satisfying scalar curvature second fundamental form sectional curvature shape operator spacelike spacetime subspace symmetric Theorem totally geodesic totally geodesic ﬁbers totally umbilical warped product