We prove a family of Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation, sharp eigenvalue estimates and a sub-Riemannian Bonnet-Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

Metabolic alterations including postprandial hyperglycemia have been implicated in the development of obesity-related diseases. Xylose is a sucrase inhibitor suggested to suppress the postprandial glucose surge. The objectives of this study were to assess the inhibitory effects of two different concentrations of xylose on postprandial glucose and insulin responses and to evaluate its efficacy in the presence of other macronutrients. Randomized double-blind cross-over studies were conducted to examine the effect of D-xylose on postprandial glucose and insulin response following the oral glucose tolerance test (OGTT). In study 1, the overnight-fasted study subjects (n = 49) consumed a test sucrose solution (50 g sucrose in 130 ml water) containing 0, 5, or 7.5 g D-xylose powder. In study 2, the overnight-fasted study subjects (n = 50) consumed a test meal (50 g sucrose in a 60 g muffin and 200 ml sucrose-containing solution). The control meal provided 64.5 g of carbohydrates, 4.5 g of fat, and 10 g of protein. The xylose meal was identical to the control meal except 5 g of xylose was added to the muffin mix. In study 1, the 5 g xylose-containing solutions exhibited significantly lower area under the glucose curve (AUCg) and area under the insulin curve (AUCi) values for 0-15 min (P < 0.0001, P < 0.0001), 0-30 min (P < 0.0001, P < 0.0001), 0-45 min (P < 0.0001, P < 0.0001), 0-60 min (P < 0.0001, P < 0.0001), 0-90 min (P < 0.0001, P < 0.0001) and 0-120 min (P = 0.0071, P = 0.0016). In study 2, the test meal exhibited significantly lower AUCg and AUCi values for 0-15 min (P < 0.0001, P < 0.0001), 0-30 min (P < 0.0001, P < 0.0001), 0-45 min (P < 0.0001, P = 0.0005), 0-60 min (P = 0.0002, P = 0.0025), and 0-90 min (P = 0.0396, P = 0.0246). In conclusion, xylose showed an acute suppressive effect on the postprandial glucose and insulin surges.

By adapting some ideas of M. Ledoux [16], [17] and [19] to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in [3]. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is non negative.

We prove a lower bound for the first eigenvalue of the sub-Laplacian on subRiemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian sphere.

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