SF3670 Semi Riemannian Geometry I,

: Hans Ringström,  hansr@kth.se, rum 3623, Lindstedsv. 25, 790 66 75.

First meeting: The first meeting will take place 15:45, Friday September 14. Please come to my office at that time.

Preparations, first meeting: In preparation for the first meeting, please read the first two chapters of O'Neill's book and do the corresponding exercises (see below). During the first meeting, the participants will be asked to solve some of the exercises on the blackboard (this will serve as a basis for discussions).

Litterature: The course will be based on the book Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neill, Academic Press, Orlando (1983).

Additional references: For those who can read German, there are notes available on the homepage of However, these will not be used in the course.

Some other references that might be good to consult are:
Homework problems: One compulsory part of the course is to complete the homework problems, available in pdf form below:
Problem set 1
Problem set 2
Problem set 3
Problem set 4

Teaching: this is a reading course, and so there will be no lectures. The first two chapters should essentially be considered as background material and chapter 6 is not part of the course. The essence of the course consists of chapters 3-8 (except 6), part of chapter 9 and chapter 10. The examination will be in the form of homework exercises to be presented orally or in writing. The exact list of excercises will be specified later. Partly, the excercises will be taken from the list below but additional problems from other sources will also be used in the examination. Starting in the fall of the year 2018, the intention is to have informal meetings every other week discussing problems and so on.

Reading instructions:

Chapter 1: The first chapter consists of basic differential geometry, something which is assumed as a prerequisite. However, it is useful to read through this chapter to get familiar with the notation O'Neill uses. Furthermore, it is sometimes convenient to keep the perspective of Einstein in mind, for whom the concept of a manifold was not of central importance. For a physicist, there are two things that make sense: 1) to set up a coordinate system and to make measurements with respect to it, 2) to compare with the measurements made by other observers. From a mathematical point view, this corresponds to local coordinates and transformation laws when you change local coordinates. The concept of a manifold, independent of an observer (i.e. a local coordinate system) is from this point of view meaningless. It is useful to keep this in mind when reading old references, especially in General Relativity. However, from the modern mathematical perspective, the manifold is, of course, the central object.

Exercises: 8,12,14,16.

Chapter 2: The material of chapter 2 is also essentially elementary, but again, it is necessary to get familiar with the notation and the material that was not covered in earlier courses, see for example the last three sections of the chapter.

Exercises: 6,10,12,13,14.

Chapter 3: In this chapter the basic geometric concepts are introduced. To start with, the concept of a metric and of a Semi-Riemannian manifold are defined. The basic model on which the concept of a Riemannian metric and of a Riemannian manifold are based is n-dimensional Euclidean space with the ordinary Euclidean metric. In the case of Lorentz geometry, the basic model is that of Minkowski space from Special Relativity. Given a metric, there is a canonically associated Levi-Civita connection, which allows one to differentiate a vectorfield with respect to a vector. Furthermore, the concepts parallel translation and geodesics are defined. In Riemannian geometry, geodesics should be thought of (at least locally) as length minimizing curves. In the case of Euclidean space, the geodesics are straight lines. In Lorentz geometry, there are three different types of geodesics; timelike, null and spacelike. In General Relativity, a timelike geodesic is interpreted as curve followed by a freely falling test particle and a null geodesic as the curve followed by a light ray. In Minkowski space, geodesics are straight lines, just as in the Euclidean case. Associated with the concept of a geodesic is the concept of an exponential map. This map yields special coordinates at a point that are often very convenient to use in calculations. Moreover, from the perspective of General Relativity, they correspond to a local observer in free fall.

On a formal level, the curvature associated with the Levi-Civita connection quantifies the lack of commutation of derivatives. However, there is also an important interpretetation in terms of a one parameter family of geodesics. If the setting is Lorentzian and the geodesics are timelike, the family can be thought of as a family of freely falling test particles. An infinitesimal version of the relative position of members of the family is then given by a so called Jacobi field (this concept is only introduced in chapter 8, definition 2). The second derivative of the Jacobi field intuitively represents the relative acceleration of the freely falling test particles, and this can be expressed in terms of the curvature. This connects curvature with relative acceleration of freely falling test particles, and thus with gravity.

After the Riemann curvature tensor is defined, sectional curvature, Ricci curvature, some differential operators such as grad are defined. All these concepts are of essential importance. The Ricci tensor is an essential ingredient of the Einstein equations, and Corollary 54 establishes a relationship which is crucial to Einstein's equations making sense. In Riemannian geometry, there are results where assumptions are made concerning the Ricci tensor and conclusions are obtained concerning the global topology of the underlying manifold. One example is Myer's Theorem, Theorem 24, p. 279. Another is the result of Hamilton that a connected, simply connected and complete Riemannian manifold with positive Ricci curvature is diffeomorphic to the 3-sphere (this result is, however, far beyond the scope of the present course).

Exercises: 2,4,5,6,7,8,9,10,12,17,20,21.

Chapter 4: This chapter treats the concept of a submanifold from a geometric point of view. Again, the concepts are fundamental. One obtains an induced metric, connection, second fundamental form etc. This makes it possible to relate the geometry of the submanifold with the geometry of the manifold. Some basic examples are hyperquadrics, a family to which the spheres and the hyperbolic spaces belong. The Gauss and Codazzi equations, which are of central importance when relating the geometry of the submanifold with the geometry of the manifold, are also derived in this chapter.

Exercises: 1,2,3,4,5,6,12.

Chapter 5: This chapter contains the Gauss lemma, the concept of a convex set, the local aspects of geodesics (such as the local length minimizing property for geodesics in Riemannian geometry). All of these concepts are important for local considerations, and some of the lemmas are technically more difficult to prove than one might naively expect. Nevertheless, it is important to familiarize oneself with the techniques involved. One important result is the Hopf-Rinow theorem which gives different characterizations of completeness for a Riemannian manifold. It is of interest to note that this theorem, which is not very complicated, has no analogue in the Lorentz setting. In fact, in Riemannian geometry, it is quite natural to demand that a manifold be geodesically complete. The corresponding question is of central importance also in the context of Lorentz geometry due to the interpretation of null and timelike geodesics. If a timelike geodesic is incomplete, it means that it reaches the boundary of spacetime within finite proper time. In other words, something goes dramatically wrong within a time frame the observer might survive. This has led researchers in the General Relativity community to define a singularity in terms of incomplete geodesics; if there is an incomplete null or timelike geodesic in a Lorentz manifold, there is a singularity. Following the incomplete geodesic towards its end means going into the singularity. It should be mentioned that, for this to make sense, the Lorentz manifold has to be maximal in some natural sense, since it is of course enough to remove a point in order to get incompleteness. As a consequence, the question of geodesic completness is of central importance in Lorentz geometry. Nevertheless, there are only very partial results describing the situations in which one has completeness/incompleteness. In particular, the singularity theorems by Hawking and Penrose represent some quite special conditions that ensure incompleteness. This illustrates that, even though the basic constructions in Riemann and Lorentz geometry are the same, there are quite fundamental differences, and when it comes to more subtle questions, there need not be any analogies between the two cases.

Exercises: 1,2,3,4,6,10,11,12,14,15,16.

Chapter 7: This chapter contains a lot of important constructions; Semi-Riemannian covering spaces, orbit manifolds, the volume element, normal neighbourhoods of a manifold, warped products. Constructing orbit manifolds using free and properly discontinuous subgroups of the isometry group is a standard method for obtaining new Semi-Riemannian manifolds. Many basic examples of Semi-Riemannian manifolds are warped products (for example all the basic examples of cosmological models), and it is very convenient to derive formulas for the curvature of such constructions expressed in terms of the curvature of the constitutents. When doing computations in practice, the formulas derived in this chapter occur again and again.

Exercises: 1,2,3,4,5,6,7,11,13.

Chapter 8: This chapter explains the notion of a Jacobi field. The importance of this concept in General Relativity has already been noted. An important result contained in this chapter is that complete, connected, simply connected semi-Riemannian manifolds of the same constant curvature are isometric. In Riemannian geometry, the basic examples of constant curvature are the Euclidean metric (curvature zero), the spherical metric (positive curvature) and the hyperbolic metric (negative curvature). The above mentioned results gives a characterization of these models in terms of curvature (in combination with some global properties). This result is also of interest in connection with cosmology. There are philosophical predjudices saying that our universe should, at one instant in time, be homogeneous and isotropic. This leads to the conclusion that the hypersurfaces of constant time should have constant curvature. Combining this observation with the above theorem, we obtain the conclusion that there are only the three above mentioned possibilities as far as the geometry is concerned.

Exercises: 1,2,3,4,6,7,8.