A Comprehensive Introduction to Sub-Riemannian Geometry

Abstract

and several summer schools, in the period 2008-2018. It contains material for an introductory course in sub-Remannan geometry at master or PhD level, as well as material for a more advanced course. The book attempts to be as elementary as possible but, although the main concepts are recalled, it requires a certain ability in managing object in differential geometry (vector fields, differential forms, symplectic manifolds, etc.). We try to avoid as much as possible the use of functional analysis (some is required starting from Chapter 6). We do not require any knowledge in Riemannian geometry. Actually from the book one can extract an introductory course in Riemannian geometry as a special case of sub-Riemannian one, starting from the geometry of surfaces in Chapter 1. There are few other books of sub-Riemannian geometry available. Besides the pioneering book edited by A. Bellaïche and J.-J. Risler [BR96], a nowadays classical reference is the book of R. Montgomery [Mon02], that inspired several of our chapters. More recent books, written in a language similar to the one we use, are those of F. Jean [Jea14] and L. Rifford [Rif14]; see also the collection of lectures notes [BBS16a, BBS16b]. Other related books, although with a different approach, are the monographs [BLU07] and [CDPT07]. Example of an introductory course of sub-Riemannan geometry. Chapters 2, 3 (without the appendices), 4, 7 (without 7.1), 9, 13, 21. Example of an advanced course of sub-Riemannan geometry.

Density of the signature process of fBm

“…Let {e 1 , ..., e d } be the standard basis of R d , and recall n = dim G N (R d ). By Lemma 3.32 of [1], there exists e i 1 , ..., e in ands 1 , ...,s n ∈ R such that the map φ : R n → G N (R d ), φ(s 1 , ..., s n ) = e s 1 e i 1 ⊗ · · · ⊗ e sne in , is non-degenerate ats = (s 1 , ...,s n ).…”

“…Note that p > N and let π p N : G ⌊p⌋ (R d )) → G N (R d )) be the canonical projection. It is then clear that Ψ(·) 1…”

“…Lower bound: We first prove (6.5). To this aim, for any u ∈ R n , fix an arbitrary η > 0 and let h ∈H be such that Ψ(h) 1…”

Periodic Controls in Step 2 Strictly Convex Sub-Finsler Problems

“…We consider linear-in-controls time-optimal left-invariant problems on step 2 Carnot groups, with a strictly convex control set. In particular, this class of problems contains sub-Riemannian [1][2][3] and sub-Finsler [4][5][6][7][8] problems. Our aim is to characterize extremal controls.…”

A dynamical approach to the Sard problem in Carnot groups

“…Remember that a Carnot group G of rank r and step s is a connected, simply connected and nilpotent Lie group whose Lie algebra g, here identified with the tangent at the group identity e, admits a stratification of the form: g = g 1 ⊕ · · · ⊕ g s , with g i+1 = [g, g i ] for 1 ≤ i ≤ s − 1, [g, g s ] = {0} and dim(g 1 ) = r. A Carnot group can be naturally endowed with a sub-Riemannian structure by declaring the first layer g 1 of the Lie algebra to be the horizontal space. Actually, Carnot groups are infinitesimal models for sub-Riemannian manifolds (that we do not introduce here, see [4,24,29,31]). Denoting by L g the left-translation on G by an element g ∈ G, we consider the endpoint map (1.1) F e : L 1 ([0, 1], g 1 ) → G,…”