Comments on the lectures

Lecture 1: Notes Download Notes

. Definitions of Differentiable manifolds and tangent vectors up to p. 20 of the lecture notes by Bär. Read this, and in particular read and work through the examples carefully.

Lecture 2: Notes Download Notes

. Tangent vectors as derivations and vector fields, Sections 1.4 and 1.5 in Bär. It is a very good exercise to understand the construction of the differentiable structure on the tangent bundle TM. Also think about the comment after Thm, 1.3.13 in Bär, why is the proof of the chain rule so simple? We further had a quick look on the Lie bracket of vector fields, see Chap. 8 p. 185 in Lee. We will return to this in Module 5 of the course.

(In the lectures I mentioned there are (at least) four ways of defining tangent vectors. First as equivalence classes of curves, next as derivations at a point, and third as component vectors in charts satisfying the change of coordinate rule (1.6) in Bär, see discussion p. 30. The fourth way would be as the Zariski tangent space Links to an external site., which is more common in the setting of algebraic geometry Links to an external site..)

Exercise Session 1: Notes. Download Notes.

(for a better resolution, the download is required) Stereographic and Cylindrical coordinates on the cylinder. We prove that they give the same atlas. We covered also tangent vectors and vector fields on the cylinder. Lastly, we computed the Lie bracket in two different cases and a geometric intuition was given that supports these computations.

Lecture 3: Notes Download Notes

. Definitions of Semi-Riemannian metrics and the Levi-Civita connection, Sections 2.1-2.3 up to definition 2.3.3. In this definition of the Levi-Civita connection $LaTeX: \nabla$

$\nabla$ , the torsion-free property vi) can be equivalently be formulated as $LaTeX: \nabla_X Y - \nabla_YX = [X,Y]$

$\nabla_X Y - \nabla_YX = [X,Y]$ for all vector fields LaTeX: X, Y

$X, Y$ . Using this alternative version of torsion-freeness we derived the Koszul formula Links to an external site., which proves existence and uniqueness of $LaTeX: \nabla$

$\nabla$ , and gives a tool to compute it. It is a good exercise to redo the derivation of the Koszul formula. It is easy to find a Riemannian metric on the domain LaTeX: U

$U$ of a chart $LaTeX: x: U \to V \subset \mathbb{R}^n$

$x: U \to V \subset \mathbb{R}^n$ in a differentiable manifold LaTeX: M

$M$ , one can simply take the pull-back by LaTeX: x

$x$ of the standard scalar product on $LaTeX: \mathbb{R}^n$

$\mathbb{R}^n$ . A Riemannian metric on the whole of LaTeX: M

$M$ can then be constructed by adding such local metrics using a partition of unity Links to an external site.. For a detailed construction, see pp 40-44 in the book by Lee.

Lecture 4: Notes Download Notes

. We covered most of Sections 2.4-2.6 in the lecture notes. I did not have time discuss the exponential map and normal coordinates, see pp 72-77. The exponential map is the map sending a tangent vector $LaTeX: \xi \in T_p M$

$\xi \in T_p M$ to the point LaTeX: c(1)

$c(1)$ on the unique geodesic LaTeX: c

$c$ with $LaTeX: \dot{c}(0) = \xi$

$\dot{c}(0) = \xi$ . This map is a diffeomorphism from a neighbourhood of LaTeX: 0

$0$ in

$T_pM$ to a neighbourhood of LaTeX: p

$p$ in

$M$ , and the normal coordinates at LaTeX: p

$p$ are defined using this diffeomorphism. Out of all the coordinate systems available in the differentiable structure on LaTeX: M

$M$ , the normal coordinates at LaTeX: p

$p$ are optimally adapted to the semi-riemannian metric at LaTeX: p

$p$ . In these coordinates, the metric is diagonal and the Christoffel symbols vanish at LaTeX: p

$p$ , see Prop 2.6.31. In normal coordinates, the Taylor series expansion of the semi-riemannian metric has no first order terms, see Cor. 2.6.33. In general, there is no way to choose coordinates so that the second order terms vanish as well, there the curvature appears.

Exercise Session 2: Notes. We solved several exercises in which we compute on specific examples: change of charts in metrics, isometries, Christoffel symbols of a manifold, parallel transport of a vector along a curve and geodesics. We also solved exercise (11) from the Exercise list.

Lecture 5: Notes Download Notes

. We covered the material 3.1-3.3 and part of 3.4. It's very important to calculate the curvature of specific metrics, see the exercise sheets. Please also verify the formula appearing in Remark 3.16. It is important to read all the proofs in detail. In particular, it is important that you understand, in detail, the calculation carried out at the top of page 98.

Lecture 6: Notes Download Notes

. We covered the material in Sections 3.4 and 4.1.

Exercise Session 3: N Download Notes Download otes. We solved several exercises in which we compute on specific examples: the Riemannian curvature tensor, sectional curvature, geodesic variations and Jacobi fields. (Since I didn't have enough time to cover everything, I skipped some parts during the session. But all the details were added after.)

Lecture 7: Notes Download Notes

. The lecture covered Section 4.2 and parts of Section 4.3. Read the examples in the book. Some of the proofs in the lecture were different from the proofs in the lecture notes. For that reason, it's important to also read the proofs in the lecture notes in detail.

Lecture 8: Notes Download Notes

. The lecture covered the rest of Section 4.3 and most of Section 4.4. The remainder of Section 4.4 is left to self study, but please read it carefully. In Riemannian geometry, the n-sphere and n-dimensional hyperbolic space are extremely important. They are special cases of the constructions of Section 4.4, and the basic properties of these spaces are sorted out in this section. Anti de Sitter space and de Sitter space are, again, special cases of the constructions, and they are extremely important in Lorentzian geometry and general relativity. In particular, the geometry of de Sitter space is expected to correspond to the asymptotic behaviour of the currently preferred models of the universe. Section 4.5 is optional (it is not part of the course).

Exercise Session 4: Notes Download Notes

. We solved some exercises where: we check if a subset is a submanifold (using the definition and regular values), we compute the second fundamental form, we check if a submanifold is totally geodesic, we use the curvature formula (regarding the second fundamental form), we check some properties of the gradient and the hessian.

Lecture 9: Notes. Download Notes.

In this lecture we tried to cover several concepts around vector fields and their flows, mainly from the book by Lee, but also from the book by Petersen.

We first looked at the Lie bracket (Lee p185-189). We then had a very quick look at Lie groups (Lee p150-161), important examples are the orthogonal group and the special orthogonal group (Lee Ex 7.27, 7.28, p166-167). On a Lie group, the set of left-invariant vector fields is invariant under the Lie bracket. This is a finite-dimensional vector space which can be identified with the tangent space at the identity element. With the Lie bracket it forms the Lie algebra of the Lie group (Lee p189-199). Important examples are the Lie algebra of the general linear group (Prop 8.41 p193) and the Lie algebra of the orthogonal group (Ex 8.47 p197).

Next, we looked at flows of vector fields (Lee p205-220). The flow is defined by solving ordinary differential equations for integral curves. The flow can be used to define coordinates adapted to a vector field, at a regular point of a vector field there are coordinates in which the vector field is a coordinate vector field (Thm 9.22).

The Lie derivative is defined using the flow of a vector field (Lee p227-231). The Lie derivative of vector fields turns out to be the same as the Lie bracket (Thm 9.38 p229), and is thus fundamentally different from the covariant derivative.

Vector fields commute if their Lie bracket is zero (Lee p231-236). This holds if and only if their flows commute (Thm 9.44 p233). Coordinate vector fields commute, and the converse holds as well, any commuting vector fields are coordinate vector fields. (Thm 9.46 p234).

The Lie derivative can be defined for all types of tensor fields (Petersen Sec 2.1.2) . We will in particular be interested in the Lie derivative of Semi-Riemannian metrics, which are symmetric (0,2)-tensors (Petersen Prop 2.1.2 with k=2).

Lecture 10: Notes. Download Notes.

First, we discussed smooth distributions following Lee (Chap 19, p490-501). A distributions is a smooth selection of dim=k subspaces the tangent bundle. It is is integrable if each point is contained in a k-dimensional submanifold whose tangent spaces are given by the distribution. The Frobenius theorem gives (Thm 19.12 p497) gives necessary and sufficient conditions for a distribution to be integrable. (The material on involutivity and differential form p493-496 is beyond this course at the moment, but could be understood after Module 7).

The main topic of the lecture was Killing vector fields, folllowing the book by Petersen (Chap 8.1-8.2, p313-320). Killing fields are vector fields whose flows give isometries of a (semi-)Riemannian metric. Killing vector fields are rigid objects. For a compact manifold of negative Ricci curvature there are no Killing vector fields (Bochners Theorem, Thm 8.2.2 p319). The central part of the proof is Prop 8.2.1(3) combined with the maximum principle (Ex 2.5.27 p75, this exercise is also beyond the course at the moment, but can be understood in Module 7).

Exercise Session 5: Notes. Download Notes.

We solved some exercises about: Computing integral curves and flows; Checking by definition if a vector field is Killing, Proprieties of Killing vector fields; Distributions and Frobenius Theorem.

Lecture 11: Notes Download Notes

. We covered half of chapter 5 in Christian Bär's notes. Note that the question of geodesic completeness and incompleteness is of central importance in Lorentzian geometry. However, proving the corresponding results is substantially more difficult. On the other hand, doing so leads to the so-called singularity theorems of Hawking and Penrose. In the Lorentzian setting, it is also natural to maximize the lengths of causal curves, but it is not natural to minimize length.

Lecture 12: Notes Download Notes

. We covered the second half of chapter 5. In particular, we proved the Bonnet-Myers Theorem. Note that, given the assumptions of the theorem, one also obtains the conclusion that the fundamental group is finite. There is a related result in the case of 3-dimensions: a 3-dimensional manifold satisfying the conditions is topologically a 3-sphere or a quotient of a 3-sphere. This result is due to Richard Hamilton and is substantially harder to prove.

Exercise Session 6: Notes. Download Notes.

In this session, it was only solved (partially) the first two exercises. Nevertheless, you can find in the link all planned exercises solved. These are about: Distance function, Complete Manifolds, Second Variation formula, Hopf-Rinow Theorem and finally Bonnet-Myers Theorem.

Lecture 13: Notes Download Notes

. Continuation of Riemannian geometry, following the book by Gallot-Hulin-Lafontaine. We began by looking at the fundamental group and (universal) covering spaces Links to an external site.. Using this, the Bonnet-Myers theorem tells us that the universal covering of a compact manifold with positive Ricci must be compact, and the fundamental group is finite (3.G.1). Contrary to this, the Cartan-Hadamard theorem (3.G.2) tells us that the universal covering of a compact manifold with non-positive curvature is diffeomorphic to $LaTeX: \mathbb{R}^n$

$\mathbb{R}^n$ . Our next goal is to relate curvature and volumes. For this we first need to define integrals and volumes on Riemannian manifolds, see 3.H.1-3.H.2 where the Riemannian density is defined. Then we want to compute integrals in normal coordinates using the exponential map at a point $LaTeX: m \in M$

$m \in M$ . For this we first need to understand which domain of the exponential map will parametrize the full manifold. This is given by the subset $LaTeX: U_m \subset T_m M$

$U_m \subset T_m M$ (2.C.7) and the cut locus LaTeX: cut_m

$cut_m$ , see Prop 2.113. Lemma 3.96 tells us that LaTeX: cut_m

$cut_m$ has measure zero, so integrating over LaTeX: M

$M$ is the same as integrating over LaTeX: U_m

$U_m$ in normal coordinates. Finally we need to compute the Riemannian density in normal coordinates. This is done before (3.97) using the relation between Jacobi fields and the differential of the exponential map. The final formula (3.97) will be used next time!

Lecture 14: Notes Download Notes

. Final lecture on Riemannian geometry. We looked at results relating curvature and volume. First the relation between scalar curvature and volumes of small balls at a point (Galot-Hulin-Lafontaine 3.H.4 Thm 3.98). The scalar curvature appears as the first non-trivial term in the Taylor expansion of volume as a function of the radius of the ball. Next, we looked at the theorem of Bishop-Gunter (G-H-L 3.H.5 Thm 3.101), which relates the volume of balls in a manifold with a curvature bound to volumes of balls in a modelspace where the curvature bound is an equality. From this theorem follows two resultts on the fundamental group of a compact manifold with a curvature bound. The Milnor-Wolf theorem (G-H-L 3.I.1 Thm 3.106) tells us that the fundamental group of manifold with non-negative Ricci-curvature has polynomial growth. The Milnor theorem (G-H-L 3.I.2 Thm 3.110) says that the fundamental group of manifold with negative sectional curvature has exponential growth.

Exercise Session 7: Notes. Download Notes.

In this session, you can find exercises concerning: Jacobi fields and conjugate points, the Weingarten map/Shape operator and Curvature and growth of the fundamental group.

Lecture 15: Notes Download Notes

. The lecture largely covered the material on pp. 349-367 of Lee's book. Note that covariant k-tensors can be defined as multilinear maps from k copies of the underlying vector space to the real numbers. In particular, the discussion of tensors in Chapter 12 is not needed in order to understand the material of Chapter 14. Moreover, the tensor product can be defined as in the lecture notes. In the discussion of differential forms on manifolds, Lee speaks of vector bundles. This is not necessary in order to define differential forms (see the lecture notes).

Lecture 16: Notes Download Notes

. The lecture started with a discussion of orientations. Note that the change of variables formula involves the absolute value of the Jacobi matrix of the change of coordinates. However, the change of variables relation for n-forms on an n-manifold involve the determinant (without the absolute value sign). In order to keep track of the corresponding sign ambiguity when integrating n-forms on n-manifolds, it is necessary to introduce an orientation. This topic is discussed in Chapter 15 of Lee's book. The material relevant for the course is contained in pages 377-383 and the part on boundary orientations (page 386 and half of 387). Following the discussion of orientations, we defined the integral of an n-form on an oriented n-manifold and proved Stokes' Theorem. The relevant material in Lee's book is pages 400-415 (Theorem 16.17). I also recommend everyone to go through the derivation of Theorems 16.32 and 16.34, and to relate Stokes' Theorem to the results from vector calculus.

Exercise Session 8: Notes Download Notes

. In this session, you can find exercises concerning: Push-forward and pull back of tensors, Lie derivative of tensors, orientation and integration on manifolds, orientability.

Lecture 17: Notes Download Notes

. The material in this lecture is not officially part of the course, and will not be on the exam. We roughly covered Chapters 17 and 18 in Lee's book. The de Rham cohomology groups of a smoothe manifold are the vector spaces of closed forms modulo exact forms. The first deep result is their homotopy invariance, Prop 17.10 and Thm 17.11. This means that even though the de Rham cohomology groups are defined using the differentiable structure, they are invariants of the underlying topological space. With this as a starting point, a topological theory can be developed, including the Mayer-Vietoris long exact sequence (Thm 17.20), and the isomorphism to real-valued singular cohomology (Thm 18.14). In the Mayer-Vietoris sequence, the "connecting homomorphism" is usually somewhat mysterious. Working with differential forms it can be computed rather explicitly using a partition of unity (Cor. 17.42). For a beautiful treatment of algebraic topology in terms of differential forms, see Bott-Tu Links to an external site.. We ended the course by looking at the additional structure on forms and de Rham cohomology given by a Riemannian metric. In Exercise 16-18, the inner product is extended to the bundles of forms in the natural way. This, together with a choice of orientation, gives rise to the Hodge-star operator. In Exercise 16-22, the codifferential LaTeX: d^*

$d^*$ is defined using the Hodge star. This is the adjoint operator to the exterior differential. Finally, in exercise 17-22, the Laplace-Beltrami operator $LaTeX: \Delta = dd^* + d^*d$

$\Delta = dd^* + d^*d$ is defined, and it is shown that every cohomology class $LaTeX: [\omega]$

$[\omega]$ has a unique harmonic representative, that is a form with $LaTeX: \Delta \omega = 0$

$\Delta \omega = 0$ or equivalently $LaTeX: d\omega = 0$

$d\omega = 0$ and $LaTeX: d^*\omega = 0$

$d^*\omega = 0$ . Here, the existence of such a form needs the theory of elliptic PDEs, see Warner Links to an external site.. With this insight, we can use curvature to get information on cohomology. Similar to Killing vector fields, there is a Bochner formula for harmonic 1-forms, see Gallot-Hulin-Lafontaine Prop 4.36. From this follows the conclusion that if Ricci curvature is positive (or non-negative and positive somewhere), then the first de Rham cohomology group vanishes. This is similar to, but weaker than, the Bonnet-Myers theorem. The strength of the Bochner method is that can be generalized, see Chapter 9 in the book by Petersen.

Exercise Session 9: Notes. Download Notes.

In this session, we solved the rest of the exercises from the exercise list. These are about: Orientability, Symplectic forms and Cohomology group.