Introduction to riemannian and subriemannian geometry. This set of notes is divided into three chapters and two appendices. Our goal was to present the key ideas of riemannian geometry up to the generalized gaussbonnet theorem. Ricci curvature, scalar curvature, and einstein metrics 31 3. These notes on riemannian geometry use the bases bundle and frame bundle, as in geometry of manifolds, to express the geometric structures. Thus, these maps allows riemannian metrics to be defined in neighborhoods. The idea behind the exponential map is to parametrize a riemannian manifold, m, locally near any p. The metric induced by the riemannian metric is complete. Chapter 1 is concerned with the notions of totally nonholonomic distributions and subriemannian structures. The derivation of the exponential map of matrices, by g. Affine connections, geodesics, torsion and curvature, the exponential map, and the riemannian connection follow quickly. What is the idea behind the definition of an exponential.
Geodesic and exponential maps on a riemannian manifold. Actu ally from the book one can extract an introductory course in riemannian geometry as a special case of subriemannian one, starting from the geometry of surfaces in chapter 1. Semiriemannian manifolds 3 maps between manifolds 4 part 2. It starts with the definition of riemannian and semiriemannian structures on manifolds. The study of riemannian geometry is rather meaningless without some basic knowledge on gaussian geometry i.
Rn is called euclidean space of dimension n and the riemannian geometry. We introduce the basic concepts of differential geometry. Foliation of tangent bundle arising from exponential map. Generic singularities of the exponential map on riemannian. The taylor expansion of the exponential map and geometric.
African institute for mathematical sciences south africa 272,390 views 27. Regions where the exponential map at regular points of sub. Riemannian geometry and statistical machine learnin g guy lebanon cmulti05189. Some exercises are included at the end of each section to give you something to think about. Riemannian exponential map and we study conjugate points. Riemannian geometry and statistical machine learnin g. Probabilities and statistics on riemannian manifolds.
A comprehensive introduction to subriemannian geometry. It implies that any two points of a simply connected complete riemannian manifold with nonpositive sectional curvature are joined by a unique. Riemannian geometry from wikipedia, the free encyclopedia. Exponential map and cut locus from the theory of second order differential equations, we know. Riemannian manifold, geodesic maps, exponential maps. International journal of mathematics trends and technology volume 10 number 1 jun 2014. Some portion of this, or all of it, is known as the hopfrinow. Tuynman pdf lecture notes on differentiable manifolds, geometry of surfaces, etc. The taylor series for of the metric in normal coordinates is an unusual feature. In other words, a geodesic is a curve that paralleltransports its own tangent vector. In particular the leftinvariant elds integrate out to geodesics. Introduction as a preliminary step to understand the global geometry of a riemannian hilbert manifold m, one studies singularities of its exponential map. Exponential map of a weak riemannian hilbert manifold article pdf available in illinois journal of mathematics 484 march 2004 with 51 reads how we measure reads. Furthermore, when the order is one, we obtain a new case which also applies to finitedimensional manifolds and.
Chapter 6 continues the study of geodesics, focusing on their distanceminimizing properties. In particular, we construct normal forms for the l 2 riemannian exponential map near all regular conjugate points of order greater than one. Chapter 7 geodesics on riemannian manifolds upenn cis. Riemannian geometry is the branch of differential geometry that studies riemannian manifolds, smooth manifolds with a riemannian metric, i. International journal of mathematics trends and technology. It has more problems and omits the background material. Rescaling lemma of geodesics, geodesic flow as dynamical system, acting on the tangent bundle, identification of the tangent space at v of tm with the cartesian product of two copies of the tangent spaces of m at the footpoint of v, horizontal and vertical subspaces of this tangent space, definition of the riemannian exponential map. This gives, in particular, local notions of angle, length of curves, surface area and volume. Riemannian geometry studies smooth manifolds endowed with a smoothly changing metric. The rst chapter provides the foundational results for riemannian geometry. Pdf exponential map of a weak riemannian hilbert manifold. Thus the exponential map from lie group theory is the same as the exponential map of riemannian geometry. There are few other books of subriemannian geometry available.
In chapter 9, we consider twodimensional subriemannian metrics. For a kstep subriemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. Sobolev metrics on the manifold of all riemannian metrics bauer, martin, harms, philipp, and michor, peter w. We do not require any knowledge in riemannian geometry. Every point possesses a totally normal neighbourhood. In riemannian geometry, an exponential map is a map from a subset of a tangent space tpm of a riemannian manifold or pseudoriemannian manifold m to. The rst result on fredholmness of a riemannian exponential map was proved in mi2 under the assumption. Riemannian geometry university of helsinki confluence. The exponential and logarithmic maps take geodesics in the neighborhood of a point on a riemannian manifold to that points tangent space, where unit vectors can be defined. Introduction to differential and riemannian geometry. A geometric understanding of ricci curvature in the. Generic singularities of the exponential map on riemannian manifolds fopke klok 1 geometriae dedicata volume 14, pages 317 342 1983 cite this article. Affine connections, geodesics, torsion and curvature, the exponential map, and the riemannian connection. Then locally around each point the exponential map is a bilipschitz homeomorphism.
From those, some other global quantities can be derived by. A note on k potence preservers on matrix spaces over complex field song, xiaofei, cao, chongguang, and zheng. These notes cover the basics of riemannian geometry, lie groups, and symmetric. Connections on submanifolds and pullback connections 19 7. Eratosthenes measurement but cited strabo 63bc 23bc and ptolomy 100ac 170ac, who wrongly computed 29000km instead of 40000km. However, from a computational point of view, we have to restrict the mea. If we take the riemannian metric on gto be the biinvariant metric, then exp e coincides with the exponential map exp. Given a riemannian manifold, the exponential map at a point is a function acting on a vector in the tangent space at that point defined using a constant speed geodesic originating at that point.
Lecture 1 introduction to riemannian geometry, curvature. These statements, in turn, imply that any two points of m can be connected by a geodesic. The main goal of these lectures is to give an introduction to subriemannian geometry and optimal transport, and to present some of the recent progress in these two elds. According to the smooth dependence in ode theory, the. Ive read in several books, including milnors morse theory and petersens riemannian geometry, that the exponential map in riemannian geometry is named so because it agrees with the exponential map in lie theory, at least for a certain choice of metric on the lie group is this the real reason why riemannian geometers originally called the exponential map by that name. Normal forms for the l2 riemannian exponential map on. First, some elementary ideas from the calculus of variations are introduced. We study the conjugate locus of the group of volumepreserving diffeomorphisms of a twodimensional manifold.
M in terms of a map from the tangent space tpm to the. He was among many other things a cartographer and many terms in modern di erential geometry chart, atlas, map, coordinate system, geodesic, etc. Matrices m 2c2 are unitary if mtm idand special if detm 1. I keep seeing references about some type of jacobifield thing but i cant seem to find anything expressing both the first and. How did the exponential map of riemannian geometry get its. Terse notes on riemannian geometry tom fletcher january 26, 2010 these notes cover the basics of riemannian geometry, lie groups, and symmetric spaces. The derivative of the exponential map can be expressed in terms of jacobi fields. You will find a proof in any book on riemannian geometry.