Notes on differential geometry and lie groups

Notes on Differential Geometry and Lie Groups Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: jean@cis.upenn.edu c Jean Gallier Please, not reproduce without permission of the authors August 14, 2016 To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie Preface The motivations for writing these notes arose while I was coteaching a seminar on Special Topics in Machine Perception with Kostas Daniilidis in the Spring of 2004 In the Spring of 2005, I gave a version of my course Advanced Geometric Methods in Computer Science (CIS610), with the main goal of discussing statistics on diffusion tensors and shape statistics in medical imaging This is when I realized that it was necessary to cover some material on Riemannian geometry but I ran out of time after presenting Lie groups and never got around to doing it! Then, in the Fall of 2006 I went on a wonderful and very productive sabbatical year in Nicholas Ayache’s group (ACSEPIOS) at INRIA Sophia Antipolis, where I learned about the beautiful and exciting work of Vincent Arsigny, Olivier Clatz, Herv´e Delingette, Pierre Fillard, Gr´egoire Malandin, Xavier Pennec, Maxime Sermesant, and, of course, Nicholas Ayache, on statistics on manifolds and Lie groups applied to medical imaging This inspired me to write chapters on differential geometry, and after a few additions made during Fall 2007 and Spring 2008, notably on left-invariant metrics on Lie groups, my little set of notes from 2004 had grown into the manuscript found here Let me go back to the seminar on Special Topics in Machine Perception given in 2004 The main theme of the seminar was group-theoretical methods in visual perception In particular, Kostas decided to present some exciting results from Christopher Geyer’s Ph.D thesis [76] on scene reconstruction using two parabolic catadioptric cameras (Chapters and 5) Catadioptric cameras are devices which use both mirrors (catioptric elements) and lenses (dioptric elements) to form images Catadioptric cameras have been used in computer vision and robotics to obtain a wide field of view, often greater than 180◦ , unobtainable from perspective cameras Applications of such devices include navigation, surveillance and vizualization, among others Technically, certain matrices called catadioptric fundamental matrices come up Geyer was able to give several equivalent characterizations of these matrices (see Chapter 5, Theorem 5.2) To my surprise, the Lorentz group O(3, 1) (of the theory of special relativity) comes up naturally! The set of fundamental matrices turns out to form a manifold F, and the question then arises: What is the dimension of this manifold? Knowing the answer to this question is not only theoretically important but it is also practically very significant, because it tells us what are the “degrees of freedom” of the problem Chris Geyer found an elegant and beautiful answer using some rather sophisticated concepts from the theory of group actions and Lie groups (Theorem 5.10): The space F is isomorphic to the quotient O(3, 1) × O(3, 1)/HF , where HF is the stabilizer of any element F in F Now, it is easy to determine the dimension of HF by determining the dimension of its Lie algebra, which is As dim O(3, 1) = 6, we find that dim F = · − = Of course, a certain amount of machinery is needed in order to understand how the above results are obtained: group actions, manifolds, Lie groups, homogenous spaces, Lorentz groups, etc As most computer science students, even those specialized in computer vision or robotics, are not familiar with these concepts, we thought that it would be useful to give a fairly detailed exposition of these theories During the seminar, I also used some material from my book, Gallier [73], especially from Chapters 11, 12 and 14 Readers might find it useful to read some of this material beforehand or in parallel with these notes, especially Chapter 14, which gives a more elementary introduction to Lie groups and manifolds For the reader’s convenience, I have incorporated a slightly updated version of chapter 14 from [73] as Chapters and of this manuscript In fact, during the seminar, I lectured on most of Chapter 5, but only on the “gentler” versions of Chapters 7, 9, 16, as in [73], and not at all on Chapter 28, which was written after the course had ended One feature worth pointing out is that we give a complete proof of the surjectivity of the exponential map exp : so(1, 3) → SO0 (1, 3), for the Lorentz group SO0 (3, 1) (see Section 6.2, Theorem 6.17) Although we searched the literature quite thoroughly, we did not find a proof of this specific fact (the physics books we looked at, even the most reputable ones, seem to take this fact as obvious, and there are also wrong proofs; see the Remark following Theorem 6.4) We are aware of two proofs of the surjectivity of exp : so(1, n) → SO0 (1, n) in the general case where where n is arbitrary: One due to Nishikawa [138] (1983), and an earlier one due to Marcel Riesz [146] (1957) In both cases, the proof is quite involved (40 pages or so) In the case of SO0 (1, 3), a much simpler argument can be made using the fact that ϕ : SL(2, C) → SO0 (1, 3) is surjective and that its kernel is {I, −I} (see Proposition 6.16) Actually, a proof of this fact is not easy to find in the literature either (and, beware there are wrong proofs, again see the Remark following Theorem 6.4) We have made sure to provide all the steps of the proof of the surjectivity of exp : so(1, 3) → SO0 (1, 3) For more on this subject, see the discussion in Section 6.2, after Corollary 6.13 One of the “revelations” I had while on sabbatical in Nicholas’ group was that many of the data that radiologists deal with (for instance, “diffusion tensors”) not live in Euclidean spaces, which are flat, but instead in more complicated curved spaces (Riemannian manifolds) As a consequence, even a notion as simple as the average of a set of data does not make sense in such spaces Similarly, it is not clear how to define the covariance matrix of a random vector Pennec [140], among others, introduced a framework based on Riemannian Geometry for defining some basic statistical notions on curved spaces and gave some algorithmic methods to compute these basic notions Based on work in Vincent Arsigny’s Ph.D thesis, Arsigny, Fillard, Pennec and Ayache [8] introduced a new Lie group structure on the space of symmetric positive definite matrices, which allowed them to transfer strandard statistical concepts to this space (abusively called “tensors.”) One of my goals in writing these notes is to provide a rather thorough background in differential geometry so that one will then be well prepared to read the above papers by Arsigny, Fillard, Pennec, Ayache and others, on statistics on manifolds At first, when I was writing these notes, I felt that it was important to supply most proofs However, when I reached manifolds and differential geometry concepts, such as connections, geodesics and curvature, I realized that how formidable a task it was! Since there are lots of very good book on differential geometry, not without regrets, I decided that it was best to try to “demistify” concepts rather than fill many pages with proofs However, when omitting a proof, I give precise pointers to the literature In some cases where the proofs are really beautiful, as in the Theorem of Hopf and Rinow, Myers’ Theorem or the Cartan-Hadamard Theorem, I could not resist to supply complete proofs! Experienced differential geometers may be surprised and perhaps even irritated by my selection of topics I beg their forgiveness! Primarily, I have included topics that I felt would be useful for my purposes, and thus, I have omitted some topics found in all respectable differential geomety book (such as spaces of constant curvature) On the other hand, I have occasionally included topics because I found them particularly beautiful (such as characteristic classes), even though they not seem to be of any use in medical imaging or computer vision In the past two years, I have also come to realize that Lie groups and homogeneous manifolds, especially naturally reductive ones, are two of the most important topics for their role in applications It is remarkable that most familiar spaces, spheres, projective spaces, Grassmannian and Stiefel manifolds, symmetric positive definite matrices, are naturally reductive manifolds Remarkably, they all arise from some suitable action of the rotation group SO(n), a Lie group, who emerges as the master player The machinery of naturaly reductive manifolds, and of symmetric spaces (which are even nicer!), makes it possible to compute explicitly in terms of matrices all the notions from differential geometry (Riemannian metrics, geodesics, etc.) that are needed to generalize optimization methods to Riemannian manifolds The interplay between Lie groups, manifolds, and analysis, yields a particularly effective tool I tried to explain in some detail how these theories all come together to yield such a beautiful and useful tool I also hope that readers with a more modest background will not be put off by the level of abstraction in some of the chapters, and instead will be inspired to read more about these concepts, including fibre bundles! I have also included chapters that present material having significant practical applications These include Chapter 8, on constructing manifolds from gluing data, has applications to surface reconstruction from 3D meshes, Chapter 20, on homogeneous reductive spaces and symmetric spaces, has applications to robotics, machine learning, and computer vision For example, Stiefel and Grassmannian manifolds come up naturally Furthermore, in these manifolds, it is possible to compute explicitly geodesics, Riemannian distances, gradients and Hessians This makes it possible to actually extend optimization methods such as gradient descent and Newton’s method to these manifolds A very good source on these topics is Absil, Mahony and Sepulchre [2] Chapter 19, on the “Log-Euclidean framework,” has applications in medical imaging Chapter 26, on spherical harmonics, has applications in computer graphics and computer vision Section 27.1 of Chapter 27 has applications to optimization techniques on matrix manifolds Chapter 30, on Clifford algebras and spinnors, has applications in robotics and computer graphics Of course, as anyone who attempts to write about differential geometry and Lie groups, I faced the dilemma of including or not including a chapter on differential forms Given that our intented audience probably knows very little about them, I decided to provide a fairly detailed treatment, including a brief treatment of vector-valued differential forms Of course, this made it necessary to review tensor products, exterior powers, etc., and I have included a rather extensive chapter on this material I must aknowledge my debt to two of my main sources of inspiration: Berger’s Panoramic View of Riemannian Geometry [19] and Milnor’s Morse Theory [126] In my opinion, Milnor’s book is still one of the best references on basic differential geometry His exposition is remarkably clear and insightful, and his treatment of the variational approach to geodesics is unsurpassed We borrowed heavily from Milnor [126] Since Milnor’s book is typeset in “ancient” typewritten format (1973!), readers might enjoy reading parts of it typeset in LATEX I hope that the readers of these notes will be well prepared to read standard differential geometry texts such as Carmo [60], Gallot, Hulin, Lafontaine [74] and O’Neill [139], but also more advanced sources such as Sakai [152], Petersen [141], Jost [100], Knapp [107], and of course Milnor [126] The chapters or sections marked with the symbol contain material that is typically more specialized or more advanced, and they can be omitted upon first (or second) reading Chapter 23 and its successors deal with more sophisticated material that requires additional technical machinery Acknowledgement: I would like to thank Eugenio Calabi, Chris Croke, Ron Donagi, David Harbater, Herman Gluck, Alexander Kirillov, Steve Shatz and Wolfgand Ziller for their encouragement, advice, inspiration and for what they taught me I also thank Kostas Daniilidis, Spyridon Leonardos, Marcelo Siqueira, and Roberto Tron for reporting typos and for helpful comments Contents The 1.1 1.2 1.3 1.4 1.5 1.6 Matrix Exponential; Some Matrix Lie Groups The Exponential Map Some Classical Lie Groups Symmetric and Other Special Matrices Exponential of Some Complex Matrices Hermitian and Other Special Matrices The Lie Group SE(n) and the Lie Algebra se(n) Basic Analysis: Review of Series and Derivatives 2.1 Series and Power Series of Matrices 2.2 The Derivative of a Function Between Normed Spaces 2.3 Linear Vector Fields and the Exponential 2.4 The Adjoint Representations A Review of Point Set Topology 3.1 Topological Spaces 3.2 Continuous Functions, Limits 3.3 Connected Sets 3.4 Compact Sets 3.5 Quotient Spaces 15 15 25 30 33 36 37 43 43 53 68 72 79 79 86 93 99 105 Introduction to Manifolds and Lie Groups 111 4.1 Introduction to Embedded Manifolds 111 4.2 Linear Lie Groups 130 4.3 Homomorphisms of Linear Lie groups and Lie Algebras 142 Groups and Group Actions 5.1 Basic Concepts of Groups 5.2 Group Actions: Part I, Definition and Examples 5.3 Group Actions: Part II, Stabilizers and Homogeneous Spaces 5.4 The Grassmann and Stiefel Manifolds 5.5 Topological Groups The Lorentz Groups 153 153 159 171 179 183 191 10 CONTENTS 6.1 6.2 6.3 6.4 6.5 The Lorentz Groups O(n, 1), SO(n, 1) and SO0 (n, 1) The Lie Algebra of the Lorentz Group SO0 (n, 1) Polar Forms for Matrices in O(p, q) Pseudo-Algebraic Groups More on the Topology of O(p, q) and SO(p, q) Manifolds, Tangent Spaces, Cotangent Spaces 7.1 Charts and Manifolds 7.2 Tangent Vectors, Tangent Spaces 7.3 Tangent Vectors as Derivations 7.4 Tangent and Cotangent Spaces Revisited 7.5 Tangent Maps 7.6 Submanifolds, Immersions, Embeddings 191 205 223 230 232 237 237 255 260 269 275 279 Construction of Manifolds From Gluing Data 285 8.1 Sets of Gluing Data for Manifolds 285 8.2 Parametric Pseudo-Manifolds 300 Vector Fields, Integral Curves, Flows 9.1 Tangent and Cotangent Bundles 9.2 Vector Fields, Lie Derivative 9.3 Integral Curves, Flows, One-Parameter Groups 9.4 Log-Euclidean Polyaffine Transformations 9.5 Fast Polyaffine Transforms 305 305 309 317 326 329 10 Partitions of Unity, Covering Maps 331 10.1 Partitions of Unity 331 10.2 Covering Maps and Universal Covering Manifolds 340 11 Riemannian Metrics, Riemannian Manifolds 349 11.1 Frames 349 11.2 Riemannian Metrics 351 12 Connections on Manifolds 12.1 Connections on Manifolds 12.2 Parallel Transport 12.3 Connections Compatible with a Metric 357 358 362 366 13 Geodesics on Riemannian Manifolds 13.1 Geodesics, Local Existence and Uniqueness 13.2 The Exponential Map 13.3 Complete Riemannian Manifolds, Hopf-Rinow, Cut 13.4 Convexity, Convexity Radius 375 376 382 391 397 Locus ... 463 16 Lie Groups, Lie Algebra, Exponential Map 16.1 Lie Groups and Lie Algebras 16.2 Left and Right Invariant Vector Fields, Exponential Map 16.3 Homomorphisms, Lie Subgroups... theory of linear differential equations with constant coefficients But most of all, as we mentioned earlier, it is a stepping stone to Lie groups and Lie algebras On the way to Lie algebras, we... manifolds and Lie groups applied to medical imaging This inspired me to write chapters on differential geometry, and after a few additions made during Fall 2007 and Spring 2008, notably on left-invariant