Notes on Differential Geometry and Lie Groups Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail:
[email protected] c Jean Gallier Please, do not reproduce without permission of the author March 25, 2009
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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 [62] on scene reconstruction using two parabolic catadioptric cameras (Chapters 4 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 3
4 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 3. As dim O(3, 1) = 6, we find that dim F = 2 · 6 − 3 = 9. 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 fai