Does differential geometry have anything to do with statistics? I am doing master in statistics and I am advised to learn differential geometry. I would be happier to hear about statistical applications for differential geometry since this would make me motivated. Does anyone happen to know applications for differential geometry in statistics?
 A: Riemannian geometry is used in the study of random fields (a generalization of stochastic processes), where the process doesn't have to be stationary. The reference I'm studying is given below with two reviews. There are applications in oceanography, astrophysics and brain imaging.
Random Fields and Geometry, Adler, R. J., Taylor, Jonathan E.
http://www.springer.com/us/book/9780387481128#otherversion=9781441923691
Reviews:
"Developing good bounds for the distribution of the suprema of a Gaussian field $f$, i.e., for the quantity $\Bbb{P}\{\sup_{t\in M}f(t)\ge u\}$, has been for a long time both a difficult and an interesting subject of research. A thorough presentation of this problem is the main goal of the book under review, as is stated by the authors in its preface. The authors develop their results in the context of smooth Gaussian fields, where the parameter spaces $M$ are Riemannian stratified manifolds, and their approach is of a geometrical nature. The book is divided into three parts. Part I is devoted to the presentation of the necessary tools of Gaussian processes and fields. Part II concisely exposes the required prerequisites of integral and differential geometry. Finally, in part III, the kernel of the book, a formula for the expectation of the Euler characteristic function of an excursion set and its approximation to the distribution of the maxima of the field, is precisely established. The book is written in an informal style, which affords a very pleasant reading. Each chapter begins with a presentation of the matters to be addressed, and the footnotes, located throughout the text, serve as an indispensable complement and many times as historical references. The authors insist on the fact that this book should not only be considered as a theoretical adventure and they recommend a second volume where they develop indispensable applications which highlight all the power of their results." (José Rafael León for Mathematical Reviews)
"This book presents the modern theory of excursion probabilities and the geometry of excursion sets for … random fields defined on manifolds. ... The book is understandable for students … with a good background in analysis. ... The interdisciplinary nature of this book, the beauty and depth of the presented mathematical theory make it an indispensable part of every mathematical library and a bookshelf of all probabilists interested in Gaussian processes, random fields and their statistical applications." (Ilya S. Molchanov, Zentralblatt MATH, Vol. 1149, 2008)
A: One area of statistics/applied mathematics where differential geometry is used in an essential way (together with a lot of other areas of mathematics!) is pattern theory.  You could have a look at the book by Ulf Grenander: https://www.amazon.com/Pattern-Theory-Representation-Inference-European/dp/0199297061/ref=asap_bc?ie=UTF8   or the somewhat more accessible text by David Mumford (a fields medal winner no less):  https://www.amazon.com/Pattern-Theory-Stochastic-Real-World-Mathematics/dp/1568815794/ref=pd_bxgy_14_img_2?_encoding=UTF8&pd_rd_i=1568815794&pd_rd_r=Q40ESHME10ZPC7XYVT59&pd_rd_w=fBcaR&pd_rd_wg=LIesY&psc=1&refRID=Q40ESHME10ZPC7XYVT59
From the preface of the last text:  

The term “pattern theory” was coined by Ulf Grenander to distinguish
  his approach to the analysis of patterned structures in the world from
  “pattern recognition.” In this book, we use it in a rather broad sense
  to include the statistical methods used in analyzing all “signals”
  generated by the world, whether they be images, sounds, written text,
  DNA or protein strings, spike trains in neurons, or time series of
  prices or weather; examples from all of these appear either
  inGrenander’s book Elements of Pattern Theory [94] or in the work of
  our colleagues, collaborators, and students on pattern theory.

One example where differential geometry is used is for face models.
Trying to answer the question (in comments) by @whuber, look at chapter 16 of Grenander's book, with title "computational anatomy". There manifolds is used to represent various parts of human anatomy (like the hearth), and diffeomorhisms used to represent changes of these anatomical manifolds, enabling comparison, modeling of growth, modeling of action of some sickness.  This ideas can be traced back to D'Arcy Thompson's monumental treatise "on growth and form" from 1917!
Grenander goes on to cite from that treatise:

In a very large part of morphology, our essential task lies in the
  comparison of related forms rather than in the precise definition of
  each; and the deformation of a complicated figure may be a phenomenon
  easy of comprehension, though the figure itself may have to be left
  unanalyzed and undefined. This process of comparison, of recognizing
  in one form a definite permutation or deformation of another, apart
  altogether from a precise and adequate understanding of the original
  “type” or standard of comparison, lies within the immediate province
  of mathematics and finds its solution in the elementary use of a
  certain method of the mathematician. This method is the Method of
  Coordinates, on which is based the Theory of Transformations.

The most well-known example of this ideas is when is some child has disappeared, say three years ago, and one publishes some photo of his face, transformed (usually using splines), into what he might look like today. 
A: Two canonical books on the subject, with reviews, then two other references:


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*Differential Geometry and Statistics, M.K. Murray, J.W. Rice

Ever since the introduction by Rao in 1945 of the Fisher information metric on a family of probability distributions there has been interest among statisticians in the application of differential geometry to statistics. This interest has increased rapidly in the last couple of decades with the work of a large number of researchers. Until now an impediment to the spread of these ideas into the wider community of statisticians is the lack of a suitable text introducing the modern co-ordinate free approach to differential geometry in a manner accessible to statisticians. This book aims to fill this gap. The authors bring to the book extensive research experience in differential geometry and its application to statistics. The book commences with the study of the simplest differential manifolds - affine spaces and their relevance to exponential families and passes into the general theory, the Fisher information metric, the Amari connection and asymptotics. It culminates in the theory of the vector bundles, principle bundles and jets and their application to the theory of strings - a topic presently at the cutting edge of research in statistics and differential geometry.


*Methods of Information Geometry, S.-I. Amari, H. Nagaoka

Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the $\alpha$-connections. The duality between the $\alpha$-connection and the $(-\alpha)$-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquitous, appearing in a variety of problems which might have no explicit relation to probability theory. Through the duality, it is possible to analyze various fundamental problems in a unified perspective. The first half of this book is devoted to a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability distributions, and the general theory of dual affine connections. The second half of the text provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book can serve as a suitable text for a topics course for advanced undergraduates and graduate students.


*Differential geometry in statistical inference, S.-I. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao, IMS Lecture Notes Monogr. Ser. Volume 10, 1987, 240 pp.

*The Role of Differential Geometry in Statistical Theory, O. E. Barndorff-Nielsen, D. R. Cox and N. Reid, International Statistical Review / Revue Internationale de Statistique, Vol. 54, No. 1 (Apr., 1986), pp. 83-96
