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Within the $\Gamma_2$-calculus of Bakry and Ledoux, we define the Riccitensor induced by a diffusion operator, we deduce precise formulas for itsbehavior under drift transformation, time change and conformal transformation,and we derive new transformation results for the curvature-dimension conditionsof Bakry-Emery as well as for those of Lott-Sturm-Villani. Our results arebased on new identities and sharp estimates for the $N$-Ricci tensor and forthe Hessian. In particular, we obtain Bochner's formula in the general setting.

Ricci Tensor for Diffusion Operators and Curvature-Dimension Inequalities under Conformal Transformations and Time Changes