Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity.

The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects.

The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making *An Introduction to Riemannian Geometry* ideal for self-study.

have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext . More information about this series at http://www.springer.com/series/223 Leonor Godinho and José Natário An Introduction to Riemannian GeometryWith Applications to Mechanics and Relativity

of the second fundamental form of associated to the frame , where and . (d)Compute the mean curvature and the Gauss curvature of . (e)Using these results, give examples of surfaces of revolution with:(1) ; (2) ; (3) ; (4) (not a plane). (Remark: Surfaces with constant zero mean curvature are called minimal surfaces ; it can be proved that if a compact surface with boundary has minimum area among all surfaces with the same boundary then it must be a minimal surface). 4.6 Notes 4.6.1

Cauchy hypersurfaces is said to be globally hyperbolic . Notice that the future and past domains of dependence of the Cauchy hypersurfaces are and . Exercise 7.10 (1) (Time-orientable double covering) Using ideas similar to those of Exercise 8.6(9) in Chap. 1, show that if is a non-time-orientable Lorentzian manifold then there exists a time-orientable double covering , i.e. a time-orientable Lorentzian manifold and a local isometry such that every point in has two preimages by . Use this to

accumulate. But since , this curve would have to intersect . Therefore must be compact. Corollary 8.7 Let be a globally hyperbolic spacetime and . Then(i) is closed; (ii) is compact. Proof Exercise 8.12(8). Proposition 8.6 is a key ingredient in establishing the following fundamental result. Theorem 8.8 Let be a globally hyperbolic spacetime with Cauchy hypersurface , and . Then, among all timelike curves connecting to , there exists a timelike curve with maximal length. This curve

Proposition 8.6 (and consequently Corollary 8.7) to the corresponding statements with compact surfaces replacing points [cf. Exercise 9.9(2)]. In particular, is closed. Therefore and so, by a straightforward generalization of Corollary 7.5, every point in this boundary can be reached from a point in by a future-directed null geodesic. Moreover, this geodesic must be orthogonal to . Indeed, at we have and so the metric takes the form If is a future-directed null geodesic with , its initial