# "Differential Geometry", Mastermath, Fall 2011

This is the web-site for the Mastermath course "Differential Geometry", Fall 2011. Here you will find all the practical informations about the course, changes that take place during the year, etc.

When, where, etc:

THE LECTURES: Wednesdays, 14:00- 16:45, in room BBL061. The course will start on week 36 (September 7) and will last until week 50 (December 14). In principle, the last 45 minutes will be alolocated to exercise classes.

LECTURERS: Marius Crainic, Florian Schaetz, Ivan Struchiner.

THE RULES FOR PASSING THE COURSE: we intend to do the following: during the semester you will receive 4 times one "not so easy problem" that you have to hand in after two weeks. The average of the marks for the hand in exercises makes on mark H. Then, at the end, there will be a short written exam, but with easy problems. For this you get another mark, call it E. The final mark will be the average (E+ H)/2, but there is the condition that both H and E have to be at least 5.

THE EXAMINATION: will be on January 18th, 2012, 15-18 in Ruppertgebouw, Zaal Wit; notes of the lectures (and summaries thereof) can be brought along

THE RESULTS of the hand in exercises can be found here: results. (The students with the last 4 numbers are not officially registered to the course.)

THE RESULTS of the examination and the corresponding total grades can be found here: results.

If you are interested in the RETAKE EXAM, contact me (Florian) via e-mail until Tuesday, February 7th. Most probably, the retake exam will take place around the end of February.

CONTENT AND LITERATURE: the course will be about Riemannian Geometry. We intend to use the following books (from which we will try to follow do Carmo more closely):

Manfred do Carmo

Riemannian geometry.

Mathematics: Theory & Applications.

Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp.

ISBN: 0-8176-3490-8

Sylvestre Gallot, Dominique Hulin and Jacques Lafontaine

Riemannian geometry.

Universitext. Springer-Verlag, Berlin, 2004. xvi+322 pp.

ISBN: 3-540-20493-8

PREREQUISITES: the standard basic notion that are tought in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a tangent vector, the tangent space (and bundle), the definition of differential forms, DeRhamd operator (and hopefully the definition of DeRham cohomology).

However, the the course will start with an "intensive reminder" on such basics (mainly an overview, without much proofs).

If you lack this bacground, it is still possible to follow the course, provided you work very hard during the first two weeks, to catch up with the basic definitions.

The schedule week by week:

WEEK 36/Lecture 1 (September 7): In this lecture we started the "intensive reminder on basic notions of Differential Geometry". This is meant to help you remember/catch up with the prerequistes. In this lecture we went through the definition of manifolds (atlasses, smooth structures, the topological conditions), some basic examples Euclidean spaces (and also opens in it, matrices, the general linear group), the sphere (viua stereographic projection), the projective space, general fibers of smooth maps (under the condition that the diferential is surjective)- itself with examples such as the sphere (again) and the orthogonal group. Then we moved to "algebraic structures" that arise: introduce the set of (real-valued) smooth functions on a manifolds- pointing out the vector space and the algebra structure; the set of vector fields on a manifold (as derivations), pointing out the vector space structure, the structure of module over the algebra of functions, as well as the structure of Lie algebra (via the Lie bracket of vector fields); the last twop structures (i.e. multiplication by functions and the Lie bracket) are related by the Leibniz identity.

WEEK 37/Lecture 2 (September 14): In this lecture we recall the notion of tangent vector (as speeds of curves, and also as "derivations at points"), the geometric description of vector fields (defined in the previous lecture algebraically), the notion of integral curves of vector fields and that of flow of vector fields. We also look at the tangent bundle and look at the notion of general vector bundles and operations with such..

WEEK 38/Lecture 3 (September 21): Finish the "Intensive reminder": more operations on vector bundles (pull-backs, k-multilinear maps and another version of the "1st theorem", antisymmetric and symmetric k-multilinear maps, differential forms (pointwise, globally and locally), wedge products, DeRham differential, DeRham cohomology, partitions of unity.

Here is the 1st hand in exercise. It should be handed in no later then October 19. If you want to send it by e-mail, please send it to: I.Struchiner@uu.nl.

WEEK 39/Lecture 4 (September 28): Metrics on vector bundles were introduced; their existence on arbitrary vector bundles was demonstrated (using partition of unity). A Riemannian manifold was defined as a manifold with a metric on its tangent bundle. Connections were introduced as operators that allow to differentiate sections along vector fields. All connections on a trivial bundle over M with fibre the vector space V were described and the existence of connections on arbitrary vector bundle established (using partition of unity). Furthermore, we introduced the vertical subspace of the tangent space of E (some vector bundle) at a point e.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 40/Lecture 5 (October 5): The non-canonical splitting of the tangent space of a vector bundle E (over M) at a point e into a vertical part -- given by the fibre -- and a horizontal part -- given by the tangent space of M at the base point of e -- was considered. The inclusion of the vertical part was described in a canonical way. Using the horizontal lift with respect to a connection on E, one also obtains a preferred inclusion of the horizontal part. Globally, this gives a splitting of the tangent bundle to E into the vertical and the horizontal subbundle (the latter depends on the connection). The pull back of a connection was defined (in terms of frames obtained by pull back). Parallel transport along a curve \gamma: I -> M was introduced by means of parallel sections (i.e. those whose derivative with respect to the connection vanish). The existence and uniqueness of the Levi-Civita connection associated to a Riemannian manifold (M,g) was proven. This is the unique connection on TM which is 1) metric with respect to g and 2) torsion-free.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 41/Lecture 6 (October 12): The local formula for the Christoffel symbols of the Levi-Civita connection was given. Geodesics were introduced as curves with 'constant' velocity. To make this precise, one sees c' as a section of the pull back bundle c*TM. As such, c' has to be parallel with respect to the pull back of the Levi-Civita connection. Local existence and uniqueness of geodesics with prescribed starting point and starting velocity was proven. The geodesic field G on TM was defined and integral curves of G were shown to be in one-to-one correspondence with geodesics on M. The exponential mapping exp was introduced and its basic properties were established.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

Here is the 2nd hand in exercise. It should be handed in no later than November 9. If you want to send it by e-mail, please send it to: florian.schaetz_at_gmail.com.

WEEK 42/Lecture 7 (October 19): Geodesic spheres and balls centered at a point p were introduced as the images of spheres/balls in the tangent space at p under the exponential map. Clearly, the geodesics inside the tangent space at p (equipped the constant inner product g(p)) that start at the origin intersect all spheres centered at 0 perpendicular. Gauss Lemma implies that this also holds in M, i.e. the geodesics starting at p intersect the geodesic spheres around p perpendicular. As a preparation of Gauss Lemma, a formula for the velocity of geodesics starting at p in terms of the exponential map was given. Moreover, properties of pull back connections were stated. Finally, Gauss Lemma was stated and proven. It roughly asserts that, applied to some vectors v, w of the tangent space to M at p, the differential of the exponential map is an isometry, i.e. preserves the inner product. To be more precise, this holds if one of the vectors is tangent to the geodesic.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 43/Lecture 8 (October 26): As an application of Gauss Lemma, we proved that geodesic spheres with center p and of sufficiently small radius intersect the geodesics starting at p perpendicular. The lenght of piecewise smooth curves was introduced and its invariance under reparametrization stated. Then we proved that any piecewise curve starting at p and ending on a geodesic sphere of sufficiently small radius r must have lenght bigger or equal to r. Moreover, the lenght is r if and only if the curve is a reparametrization of a geodesic. From this it also follows that curves which are parametrized proportional to arc length (i.e. the lenght of the curve obtained by restriction to [a,t] is proportional to (t-a)) and which minimize lenght (i.e. any other curve with the same starting and end point has bigger or equal lenght) are geodesics. Finally, the distance between two points of a connected Riemannian manifold was introduced and its basic properties were stated.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 44/Lecture 9 (November 2): After introducing basic notions concerning metric spaces, the Heine-Borel Theorem for Riemannian manifolds was proven. This Theorem gives conditions on a Riemannian manifold M which are equivalent to its completeness as a metric space (with respect to the distance). Geodesically completeness was defined and the Hopf-Rinow Theorem and some of its consequences stated. This theorem assures that a Riemannian manifold is complete as a metric space if and only if it is geodesically complete. Moreover, in this case any two points can be joined by a (possibly non-unique) geodesic of minimal length.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 45/Lecture 10 (November 9): The curvature of a Riemannian manifold was introduced and its basic properties (Bianchi identity and its behaviour under exchange of arguments) were established. The sectional curvature of a plane in a tangent space was defined. Moreover, it was shown that the sectional curvatures of all the planes of the tangent space at one point, together with the Riemannian metric at that point, is sufficient to recover the curvature at that point. Taking appropriate traces leads from the curvature to Ricci curvature and scalar curvature.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 46/Lecture 11 (November 16): The main aim of this lecture was to prove the following result: the curvature of a Riemannian manifold vanishes near a point p if and only if there is an isometry between a neighbourhood of p and an open subset of some Euclidean vector space. Along the way, the curvature of a connection on an arbitrary vector bundle E was introduced and was interpreted as the obstruction to the integrability of the horizontal subbundle of TE. To this end, Frobenius' Theorem, which states that a subbundle of the tangent bundle is integrable if and only if it is involutive, was established.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

Here is the 3rd hand in exercise. It should be handed in no later than November 30th. If you want to send it by e-mail, please send it to: florian.schaetz_at_gmail.com.

WEEK 47/Lecture 12 (November 23): Submanifolds of Riemannian manifolds were discussed. The most important structures that a submanifold obtains from the ambient manifold are: a Riemannian metric, which gives the Levi-Civita connection on the tangent bundle; the normal bundle with metric and compatible connection; the second fundamental form and the Weingarten mapping. The Weingarten equation relates the last two. There are various equations, relating components of the curvature of the ambient Riemannian manifold to the curvature of the connections on the submanifold and the second fundamental form and Weingarten mapping. As a particular case, one obtains Gauss' Theorema egregium, stating that the scalar curvature of a surface in three-dimensional Euclidean space, written in terms of the second fundamental form, is intrinsic to the surface. In the end, the energy of a piecewise smooth curve was introduced. Moreover, a relation between lenght minimizing geodesics and curves which minimize energy was proven.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 48/Lecture 13 (November 30): The energy of a one-parameter family of curves was studied. In particular, the first and second variational formulae were derived. Being a geodesic was proved to be equivalent to being a crictical point of the energy functional, meaning that the energy of any variation of the curve under consideration has a critical point at 0. In the second variational formula, the curvature appears. Two results that use this relation between curvature and energy were presented: 1) If the sectional curvature is non-positive, geodesics are local minima of energy (and lenght), i.e. any curve sufficiently close to a geodesic c and with the same starting and end points must have energy (and lenght) bigger or equal to that of c. 2) Any complete Riemannian manifold with Ricci curvature bounded away from 0 (by means of the Riemannian metric) is compact and it's fundamental group is finite. The latter result is known as the Theorem of Bonnet-Meyers.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

Here is the 4th hand in exercise. It should be handed in no later than December 20th. If you want to send it by e-mail, please send it to: florian.schaetz_at_gmail.com.

WEEK 49/Lecture 14 (December 7): The defining equation for Jacobi fields was derived by considering one-parameter families of curves all of whose members are geodsic. Same basic observations about the space of Jacobi fields along a given geodesic were made. In particular, we saw that this space forms a finite dimensional vector subspace of the space of all vector fields along the curve and it is isomorphic to two copies of the tangent space at the starting point of the geodesic. Moreover, it was proven that every Jacobi field arises from a one-parameter family of curves all of whose members are geodesic.

As an application, it was shown that the ratio of the area of a geodesic disk of radius R and the area of the disk of radius R in Euclidean space is equal to 1 up to first order (with respect to R) and deviates from 1 in second order by a term proportional to the sectional curvature. This can be interpreted as the effect that curvature has on the rate with which radial geodesics spread out in a Riemannian manifold.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.

WEEK 50/Lecture 15 (December 14): The basic facts about covering maps were recalled - in particular, the connection between the universal cover of a manifold and its fundamental group was stated. Moreover, it was proved that a local isometry between complete Riemannian manifolds is a covering map. Then conjugate points were introduced and their relation to the (differential of) the exponential map was explained. Non-positivity of the sectional curvature turned out to rule out the existence of conjugate points, i.e. given a complete Riemannian manifold of non-positive sectional curvature, the differential of the exponential map restricted to a point p is an isomorphism everywhere (on the tangent space to p).

Putting these results together, we obtained Hadamard's Theorem which says that for (M,g) a connected, complete Riemannian manifold of non-positive sectional curvature, the exponential map is a covering map. In particular, if (M,g) has in addition vanishing fundamental group, M is diffeomorphic (but in general not isometric) to some Euclidean space.

Here are some exercises related to this lecture (This is not a hand in exercise). Solutions.