We studyconformal actions of connected nilpotent lie groups on compact pseudo riemannian manifolds. I am mainly interested in large group actions on manifolds. Riemannian geometric statistics in medical image analysis is a complete reference on statistics on riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. Intrinsic polynomials for regression on riemannian manifolds. Pdf as is wellknown, the real rank of a simple lie group that acts conformally on a pseudoriemannian manifold is bounded by means of the signature. This article gives an uptodate account of the theory of discrete group actions on non riemannian homogeneous spaces. Lie group actions on manifolds 3 the isometries of a riemannian manifold form a group in an obvious manner, which we shall denote by im and call the isometry group of m.
Combining our results with results by meeks and scott invent. Compact pseudo riemannian manifolds vincent pecastaing abstract. The purpose of the meeting is to study questions from, and related to, riemannian geometry in the context of symmetry. By performing the adjoint optimization directly in the lie algebra, the computations in these spaces are greatly. We mostly aim at an audience of advanced undergraduate and graduate students. This paper constructs metrics on the space of images i defined as orbits under group actions g. This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces. We will discuss basis properties of group actions in section 3. Fundamental groups and curvature bounds, fall 20 with guofang wei. An isometry of a connected riemannian manifold is completely determined by both its value and its differential at some point. Symmetric spaces arise naturally from lie group actions on manifolds, see helgason 1978. The goal of this book is to present several central topics in geometric group. Geometric group theory is an actively developing area of mathematics.
The orbit types of g form a stratification of m and this can be used to understand the geometry of m. A few new topics have been added, notably sards theorem and transversality, a proof that infinitesimal lie group actions generate global group actions, a more thorough study of firstorder partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures. Pseudoriemannian geometry and actions of simple lie groups article in comptes rendus mathematique 3416. Riemannian geometry for the statistical analysis of. We apply an equivariant version of perelmans ricci flow with surgery to study smooth actions by finite groups on closed 3 manifolds. Sectional curvature comparison local version metric and hessian comparison jacobi elds comparison and injective raduis estimate topology of manifolds with negativenonpositive sectional curvature. We investigate the rudiments of riemannian geometry on orbit spaces.
Leftinvariant metrics and submanifold geometry abstract. The manifold can be endowed with a riemannian metric, enabling distances between shapes to be defined. Conformal actions of nilpotent groups on pseudo riemannian manifolds charles frances and karin melnick abstract. Nsf dms 1006677 \global riemannian geometry 2010 nsf dms 0204177 \global riemannian geometry 200205. The set of diffeomorphisms has many desirable mathematical properties. The goal of di erential geometry is to study the geometry and the topology of manifolds using techniques involving di erentiation in one way or another. This book provides an introduction to riemannian geometry, the geometry of curved spaces, for use in a graduate course. Lectures on geometric group theory uc davis mathematics. Apr 17, 2019 when the index is optimal and g nonexceptional, we prove that g must be conformally flat, confirming the idea that in a good dynamical context, a geometry is determined by its automorphisms group. Our main result provides a geometric splitting of the metric on such m that involves natural metrics on g. Together with a lie group action by g, m is called a gmanifold. In the case that the base space is a sphere and the abstract fiber m supports a negatively curved riemannian metric, pedro and i showed that many smooth mbundles cannot be a negatively curved bundle. Group actions are central to riemannian geometry and defining orbits control theory.
Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Our main result gives a general bound on the realrank of the lattice, which was already known for the action of the full lie group zim87b. Mg for isometric proper actions of lie groups on riemannian manifolds. The study of group actions on manifolds is the meeting ground of a variety of. Read isometric actions on pseudoriemannian nilmanifolds, annals of global analysis and geometry on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Global riemannian geometry, cuernavaca, mexico, may 2008 lie groups actions in geometry, 3 weeks of lectures, impa, august 2008 geometry and mathematical physics, in honor of the memory of krzysztof galicki, albuquerque, new mexico, october 2008 international symposium on. Applied differential geometry world scientific publishing. In differential geometry, a lie group action on a manifold m is a group action by a lie group g on m that is a differentiable map. Tatsuki seto nagoya toeplitz operators and the roehigson type index theorem. My research lies at the intersection of differential geometry and dynamical systems. Pdf lecture notes for the minicourse holonomy groups in riemannian geometry, a part of the. In particular, it is a group and also a differentiable manifold, and hence an infinitedimensional analogue of a lie group.
Group actions in riemannian geometry, chapel hill, may 2014 geometry seminar, university of oklahoma, may 20 invited speaker. A fixed point theorem of discrete group actions on. It offers a panoramic view of a selection of cuttingedge topics in differential geometry, including 4manifolds, quaternionic and octonionic geometry, twistor spaces, harmonic maps, spinors, complex and conformal geometry, homogeneous spaces and nilmanifolds, special. There are many good books covering the above topics, and we also provided our own.
It is built on the ideas and techniques from low dimensional topology, riemannian geometry, analysis, combinatorics, probability, logic and traditional group theory. Take a riemannian manifold and divide it by a group of isometries. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of riemannian geometry followed by a selection of more specialized topics. The goal of this book is to present several central topics in geometric group theory, primarily related to the large scale geometry of infinite groups and spaces on which such groups act, and to illustrate them with fundamental theorems such as gromovs theorem on groups of polynomial growth. For example, socalled free groups an a priori purely algebraic notion can be characterised geometrically via actions on trees. This is a subject with no lack of interesting examples. Our main result is that such actions on elliptic and hyperbolic 3 manifolds are conjugate to isometric actions. Hard lefschetz actions in riemannian geometry with special holonomy naichung conan leung and changzheng li abstract. The neo riemannian group and geometry extension of neo riemannian theory hindemith, fugue in e conclusion references material from this talk is from. Conformal actions of nilpotent groups on pseudoriemannian manifolds, with charles frances dvi pdf ps duke mathematical journal 153 no. Our intent is to bring together experts in the field as well as young mathematicians. Given a smooth manifold mand a lie group g, a smooth group action of gon mis a smooth mapping gm. An introduction to riemannian geometry with applications to mechanics and relativity leonor godinho and jos. With a view to dynamical systems, by keith burns and marian gidea.
The volume is a followup to the indam meeting special metrics and quaternionic geometry held in rome in november 2015. The neoriemannian group and geometry extension of neoriemannian theory hindemith, fugue in e conclusion references material from this talk is from. Group actions on spin manifolds munich personal repec archive. Riemannian symmetric spaces, and the methods for computing geodesics and distances on them, arise naturally from lie group actions on manifolds.
We investigate conformal actions of cocompact lattices in higherrank simple lie groups on compact pseudoriemannian manifolds. Our main result gives a general bound on the realrank of the lattice, which was already known for the action of the full lie group 33. Reflection groups in nonnegative curvature fang, fuquan and grove, karsten, journal of differential geometry, 2016. Topics in geometry and topology bundles, curvature, and anomalies, fall 2014 with david morrison. Compact pseudoriemannian manifolds vincent pecastaing abstract. Here the abstract fiber m is a closed smooth manifold and the structure group is diffm. When the action is not free then the natural length structure again has new features. Riemannian geometry, which enriches the text with interrelations be.
For k c or h, we write k0 r or c respectively, and we have theorem 1. Hotel to campus below is some information on taking public transportation between hotels and campus. Special metrics and group actions in geometry simon. Pdf holonomy groups in riemannian geometry researchgate. Geometry workshop, copenhagan, june 2012 invited speaker.
Newest groupactions questions mathematics stack exchange. It is shown that the topological restrictions needed to lift an action in p are more stringent than for actions in the proper poincare group p. Lie groups and geometric aspects of isometric and hamiltonian. Group actions in riemannian geometry, spring 20 with xianzhe dai.
We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. Curvature in riemannian manifolds slides, pdf local isometries, riemannian coverings and submersions, killing vector fields slides, pdf lie groups, lie algebras, and the exponential map, part ii slides, pdf metrics, connections, and curvature on lie groups slides, pdf manifolds arising from group actions slides, pdf. These topics are presented straightforward using tools from riemannian geometry, which enriches the text with interrelations between these theories and makes it shorter than usual textbooks on this. What groups g admit hamitlonian actions on closed symplectic manifolds. A purely electric spacetime is a spacetime for which there exists. Group actions on manifolds lecture notes, university of toronto. We will first introduce lie groups in terms of their action on differentiable manifolds, which is a concrete way they appear in physics, and then. Pdf introduction to riemannian manifolds download full. We studyconformal actions of connected nilpotent lie groups on compact pseudoriemannian manifolds.
Hard lefschetz actions in riemannian geometry 685 include tninvariant hyperk. We consider this group always as a topological group equipped with the compactopen topology. This course is an introduction to the theory of smooth manifolds, with an emphasis on their geometry. In particular this is applied to the poincare group p of a lorentz manifold m. There are many nontrivial examples of hamiltonian actions of compact groups. Isometric actions on pseudoriemannian nilmanifolds. If m,g is a riemannian manifold, the lie algebra xm,g xlxg 0 of. Geometry seminar max planck institute for mathematics, bonn, germany diameter of quotients of the sphere by isometric group actions november 2018 geometry oberseminar university of munster, munster, germany diameter of quotients of the sphere by isometric group actions august 2018 new trends and open problems in geometry and global analysis. Assume that two spaces, a riemannian and lorentzian, respectively, are related through a standard wickrotation. We prove that if a typep,q compact manifold m supports a conformal action of a connected nilpotent group h, then the degree of. Riemannian geometry techniques are used to give a short and constructive proof that a differentiable gfibre bundle with compact fibre is glocally trivial when g.
Lie group actions on manifolds kings college london. Notation for sets and functions, basic group theory, the symmetric group, group actions, linear groups, affine groups, projective groups, finite linear groups, abelian groups, sylow theorems and applications, solvable and nilpotent groups, pgroups, a second look, presentations of groups, building new groups from old. Alexander isaev australian national university proper group actions in complex geometry. If g is a lie group and m is a riemannian manifold, then one can study isometric actions. Principles of riemannian geometry in neural networks. It is known that the hard lefschetz action, together with k.
Lie group actions, riemannian geometry on differentiable compactifications of the hyperbolic plane and algebraic actions of sl2. The main talks are scheduled for friday afternoon, may 16, through sunday afternoon, may 18. Orbifolds generalize this notion by allowing the space to be modelled on quotients of rn by finite group actions. We investigate conformal actions of cocompact lattices in higherrank simple lie groups on compact pseudo riemannian manifolds. Conformal actions of simple lie groups on compact pseudoriemannian manifolds bader, uri. Conformal actions of nilpotent groups on pseudoriemannian manifolds charles frances and karin melnick abstract.
On topology of some riemannian manifolds of negative curvature with a compact lie group of isometries mirzaie, r. Special metrics and group actions in geometry simon george. Free groups theory books download ebooks online textbooks. Alexander isaev australian national university proper group. The current numerical implementation of the resulting geodesic pca, however, relies on bruteforce techniques that are not expected to scale gracefully.
The orbits of computational anatomy consist of anatomical shapes and medical images. Geometric group theory preliminary version under revision. Roughly speaking a manifold is a topological space locally modelled on euclidean space rn. Di erential geometry and lie groups a second course. Their main purpose is to introduce the beautiful theory of riemannian geometry, a still very active area of mathematical research. On the other hand, there is the following elementary result by delzant 1. Pdf conformal actions of simple lie groups on compact pseudo. Coauthored by the originator of the worlds leading human motion simulator human biodynamics engine, a complex, 264dof biomechanical system, modeled by differentialgeometric tools this is the first book that combines modern differential geometry with a wide spectrum of applications, from modern mechanics and physics, via.
On discontinuous group actions on nonriemannian homogeneous. Similar results hold for the euclidean group of a riemannian manifold. Riemannian geometric statistics in medical image analysis. We study the action of a connected noncompact simple lie group g on a connected finite volume pseudo riemannian manifold m. Riemannian geometry of manifolds can be proven in the more general category of orbifolds. For pseudo riemannian manifolds, isometric actions of discrete. Orbits of semisimple group actions thm hellelandhervik. Killing vector fields is finitedimensional by myerssteenrod, and by definition acts on.
Because the cat0 condition captures the essence of nonpositive curvature so well, spaces which satisfy this condition display many of the elegant features inherent in the. For a leftinvariant metric on a given lie group, we can construct a submanifold, where the ambient space is the space of all leftinvariant metrics on that lie group. In chapter 5 we develop the basic theory of proper fredholm riemannian group actions for both. For pseudoriemannian manifolds, isometric actions of discrete. Chapter 5 further develops the foundational topics for riemannian manifolds.
They are indeed the key to a good understanding of it and will therefore play a major role throughout. Ivan kol a r, jan slov ak, department of algebra and geometry faculty of science, masaryk university jan a ckovo n am 2a, cs662 95 brno, czechoslovakia. This completes earlier investigations on pseudoriemannian conformal actions of semisimple lie groups of maximal realrank 1, 10, 26. This article gives an uptodate account of the theory of discrete group actions on nonriemannian homogeneous spaces. When the index is optimal and g nonexceptional, we prove that g must be conformally flat, confirming the idea that in a good dynamical context, a geometry is determined by its automorphisms group. Pseudoriemannian geometry and actions of simple lie groups.
728 1518 202 616 58 1035 61 1296 737 1551 558 656 711 275 411 306 125 992 604 269 1422 954 1443 1009 1576 349 1121 1162 1385 816 626 69 797 118 1008 828 408 938 850 486 1144