Nicola Gigli, Universite de Nice, France

• Chapters; Prologue by Luigi Ambrosio; 1. Introduction; 2. Multiples of $\mathrm {b}$ are Kantorovich potentials; 3. The gradient flow of $\mathrm {b}$ preserves the measure; 4. The gradient flow of $\mathrm {b}$ preserves the distance; 5. The quotient space isometrically embeds into the original one; 6. ""Pythagoras' theorem"" holds; 7. The quotient space has dimension $N-1$; A. Infinitesimal Hilbertianity and behavior of gradient flows; B. Infinitesimal Hilbertianity and behavior of the distance; C. Eulerian and Lagrangian points of view on lower Ricci curvature bounds