SF3670 Semi Riemannian Geometry I,
firstname.lastname@example.org, 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:
Petersen, Peter. Riemannian geometry. Second edition.
Graduate Texts in Mathematics, 171. Springer, New York, 2006.
Sakai, Takashi. Riemannian geometry. Translations of
Mathematical Monographs, 149. American Mathematical Society, Providence, RI.
Besse, Arthur L.
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10. Springer-Verlag,
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.
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.
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.
Chapter 3: In this chapter the basic geometric concepts are introduced.
To start with, the concept of a metric and of a Semi-Riemannian
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
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).
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.
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.
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
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.