Encounter with a Geometer, Part II, Volume 47, Number 3

326 NOTICES OF THE AMS VOLUME 47, NUMBER 3 Riemannian Geometry Starting in the late 1970s, MG completely revolutionized Riemannian geometry. I mention in this section some results that reflect my taste. This article contains only a small bibliography; further references may be found in my 1998 survey article in Jahresbericht der Deutschen Mathematiker-Vereinigung.1 Except for obvious cases, every Riemannian manifold will be compact; in any case it will always be assumed complete. In (M,g) the letter M stands for the manifold and the letter g its Riemannian metric. This by definition means that at every point m of M there is an inner-product structure g( · , · ) on the tangent space TmM at this point. We begin by describing the various notions of “curvature”. The curvature tensor is the basic invariant of a Riemannian manifold. Some of its power comes from the fact it has three equivalent definitions. Two of these are in terms of the associated “LeviCivita connection”. Informally a “connection” on a smooth manifold is a way of computing directional derivatives of vector fields. These directional derivatives, which do not exist in general, will be called “covariant”. More precisely, a connection is an operator D that assigns to each pair of vector fields x and y on M a vector field Dxy on M , the covariant derivative of y with respect to x , in a fashion that is R linear in y , is C∞(M) linear in x , and satisfies Dx(fy) = x(f )Y + fDxy for all f ∈ C∞(M). The vector (Dxy)m at a point m ∈M depends only on xm and the values of y on any curve whose velocity vector at m equals xm . Consequently it is meaningful to speak of a vector field on a curve that is “parallel” along the curve: If σ is the curve and u is its tangent, then a vector field y on σ is parallel along σ if Duy = 0 on σ. If σ has domain [a, b], one knows that for each y ∈Mσ (a) there is a unique vector field Y (t) on σ such that y(a) =y and the field y(t) is parallel along σ. The passage from Mσ (a) to Mσ (t) in this way is called parallel transport. Thus a connection yields a notion of parallel transport along curves. It yields also a notion of absolute (intrinsic) derivatives of all orders for all tensors on the manifold, in particular for functions. Marcel Berger is emeritus director of research at the Centre National de la Recherche Scientifique (CNRS) and was director of the Institut des Hautes Études Scientifiques (IHÉS) from 1985 to 1994. His e-mail address is [email protected].