In this chapter we are going to construct explicitlythe full set of optimal arclength geodesics (theso-called "optimal synthesis") starting from apoint, for certain relevant sub-Riemannianstructures. To this purpose, we present a techniqueto identify the cut locus that generalizes aclassical technique used in Riemannian geometry dueto Hadamard.

In this chapter we develop a language, called"chronological calculus", that allows us to work inan efficient way with flows of nonautonomous vectorfields. The basic idea is to replace a nonlinearfinite-dimensional object, manifold $M$, with alinear infinite-dimensional object, the commutativealgebra $C^{\infty}(M)$.

In this chapter we investigate the notion of theintrinsic volume in sub-Riemannian geometry in thecase of "equiregular" structures. In particular weconsider the Popp and the Hausdorff volumes. On aRiemannian manifold these two notions coincide, butthey may be different in sub-Riemanniangeometry.

This chapter is devoted to the study of the geometricproperties of Pontryagin extremals. We then provethat small pieces of normal extremal trajectoriesare length-minimizers. To this aim, all through thischapter, we develop the language of symplecticgeometry, starting from the key concept of thePoisson bracket.

In this chapter we investigate the regularityproperties of the sub-Riemannian distance from afixed point. In particular, we prove that thesub-Riemannian distance is smooth on an open anddense subset of every compact ball, but as soon asthe distribution is not full dimensional at thepoint, every level set of the distance contains anondifferentiability point of the distanceitself.

The aim of this appendix is to describe how toconstruct canonical bundles of moving frames anddifferential invariants for parametrized curves inLagrangian Grassmannians, at least in the monotoniccase. Such curves appear as Jacobi curves ofsub-Riemannian extremals.

In this chapter we derive the sub-Riemannian heatequation and its relation to the notion of intrinsicvolume in sub-Riemannian geometry. We then discuss(without proof) the Hörmander theorem. In the lastpart of the chapter we present an elementary methodto compute the fundamental solution of the heatequation on the Heisenberg group (the celebratedGaveau–Hulanicki formula).

In this chapter we introduce the notion of anonholomonic tangent space, which can be regarded asthe “principal part” of the structure defined on amanifold by the distribution in a neighborhood of apoint. We give an intrinsic construction through thetheory of jets of curves and the notion of smoothadmissible variation. Also, we discuss the Ball–Boxtheorem, and a classification of nonholonomictangent spaces in low dimensions.

We introduce the Jacobi curve associated with a normalextremal. Heuristically, we would like to extractgeometric properties of the sub-Riemannian structureby studying the symplectic invariants of itsgeodesic flow. The simplest idea is to look forinvariants in its linearization. This subject isnaturally related to geodesic variations andgeneralizes the notion of Jacobi fields inRiemannian geometry to more general geometricstructures.

In this chapter we collect some basic definitions ofdifferential geometry, in order to recall someuseful results and to fix the notation. We assumethe reader to be familiar with the definitions ofsmooth manifold and a smooth map betweenmanifolds.

In this chapter we compute the small-time asymptoticsof the exponential map in the 3D contact case. Weshow how the structure of the cut and the conjugatelocus are encoded in these asymptotics, and weexpress them in terms of the curvatureinvariants.

In this chapter we introduce "Ehresmann connections,"with their associated notions of parallel transportand curvature. We then specify these notions in thecase of a Riemannian manifold, where one can find acanonical connection associated with the metricstructure, called Levi–Civita connection. We thenexplain how this connection is related to the theoryof Jacobi curves developed in the previouschapter.

In this chapter we study normal Pontryagin extremals onleft-invariant sub-Riemannian structures on a Liegroup $G$. Such structures provide most of theexamples in which normal Pontryagin extremals can becomputed explicitly in terms of elementaryfunctions. We introduce Lie groups by studyingsubgroups of the group of diffeomorphisms of amanifold $M$ induced by a family of vector fieldswhose Lie algebra is finite dimensional.

In this chapter we introduce the manifold of Lagrangiansubspaces of a symplectic vector space. After adescription of its geometric properties, we discusshow to define the curvature for regular curves inthe Lagrange Grassmannian. The language developed inthis chapter will be fundamental to encoding in asingle object, a curve in a space of Lagrangiansubspaces, all information concerning Jacobi fieldsalong sub-Riemannian geodesics, such as conjugatepoints and curvature.

In this chapter we motivate the study of sub-Riemmaniangeometry and we present the content of the book.

Almost-Riemannian structures form prototypes forrank-varying sub-Riemannian structures. In thischapter we study the two-dimensional case, which isvery simple since it is Riemannian almost everywherebut even so presents some interesting phenomena suchas the presence of sets of finite diameter butinfinite area. Also, the Gauss–Bonnet theorem has asurprising form in this context.

In this preliminary chapter we study the geometry ofsmooth two-dimensional surfaces in $\mathbb{R}^3$ asa “warm-up problem” and we recover some classicalresults. In the fist part of the chapter we considersurfaces in $\mathbb{R}^3$ endowed with the standardEuclidean product. In the second part we studysurfaces in the 3D pseudo-Euclidean space, that is$\mathbb{R}^3$ endowed with a sign-indefinite innerproduct.

In this chapter we show how to find certain firstintegrals, for Hamiltonian systems on Lie groups,that are automatically in involution with each otherand with the Hamiltonian. This theory will be usedto prove that the Hamiltonian system for normalPontryagin extremals for rank-2 left-invariantsub-Riemannian structures on three-dimensional Liegroups is completely integrable.