For Riemannian manifolds with a measure (M, g, e −f dvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results. IntroductionIn this paper we study smooth metric measure spaces (M n , g, e −f dvol g ), where M is a complete n-dimensional Riemannian manifold with metric g, f is a smooth real valued function on M , and dvol g is the Riemannian volume density on M . In this paper by the Bakry-Emery Ricci tensor we mean Ric f = Ric + Hessf. This is often also referred to as the ∞-Bakry-Emery Ricci Tensor. Bakry and Emery [4] extensivley studied (and generalized) this tensor and its relationship to diffusion processes. The Bakry-Emery tensor also occurs naturally in many different subjects, see e.g.[22] and [27, 1.3]. The equation Ric f = λg for some constant λ is exactly the gradient Ricci soliton equation, which plays an important role in the theory of Ricci flow. Moreover Ric f has a natural extension to metric measure spaces [21,35,36]. When f is a constant function, the Bakry-Emery Ricci tensor is the Ricci tensor so it is natural to investigate what geometric and topological results for the Ricci tensor extend to the Bakry-Emery Ricci tensor. Interestingly, Lichnerowicz [20] studied this problem at least 10 years before the work of Bakry and Emery. This has also been actively investigated recently and there are a number of interesting results in this direction which we will discuss below. Also see [8] for another modification of the Ricci tensor and Appendix A for a discussion of the N -Bakry-Emery Ricci tensor Ric N f (see (1.6) for the definition). We thank Matthew Gursky for making us aware of [20].Although there is a Bochner formula for the Bakry-Emery Ricci tensor [22] (see also (5.10)) many of the other basic geometric tools for Ricci curvature do not extend. In Section 7 we give a quick overview with examples showing that, in general, Myers' theorem, Bishop-Gromov's volume comparison, Cheeger-Gromoll's splitting theorem, and Abresch-Gromoll's excess estimate are not true for the Bakry-Emery Ricci tensor. In this paper we find conditions on f that imply versions of these theorems. In particular, we show that all these theorems hold when f is bounded. 1 These results give new tools for studying the Bakry-Emery tensor and lead to generalizations of many of * Partially supported by NSF grant DMS-0505733 1 After writing the original version of this paper, we learned that Lichnerowicz had proven the splitting theorem for f bounded. We think this result is very interesting and does not seem to be well known in the literature, so we have retained our complete proof here. Theorem 1.2 (Vol...

We define a gradient Ricci soliton to be rigid if it is a flat bundle N × ‫ޒ‬ k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-quasi Einstein equation, but we will also call it the (λ, n + m)-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some interesting new canonical examples. We also derive some new formulas involving curvatures that are analogous to those for the gradient Ricci solitons. As an application, we characterize warped product Einstein metrics when the base is locally conformally flat.

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones S n , S n 1 R and R n . This gives a new proof of the Hamilton-Ivey-Perelman classification of 3-dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of53C25