5
Pennec [143], 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 [9] introduced a new Lie group structure on the space of symmet-
ric 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 “demystify” 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 character-
istic classes), even though they do not seem to be of any use in medical imaging or computer
vision.
In the past five years, I have also come to realize that Lie groups and homogeneous man-
ifolds, 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 re-
ductive 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 naturally 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 met-
rics, 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 applica-
tions. These include
