Curvature decomposition of G_2 manifolds

Explicit formulas for the $G_2$-components of the Riemannian curvature tensor on a manifold with a $G_2$ structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact $G_2$ manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a $G_2$ structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed $G_2$ manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one $G_2$ manifolds. As a consequence every Fern\'andez-Gray type of $G_2$ structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in $G_2$.Comment: LaTeX 2e, 26 pages, 2 tables. Changes in version 2: shortened, reorganized, misprints corrected, several remarks and new introduction. A formula in the proof of Theorem 1.2a has been corrected. Submitte

Cohomogeneity-one G2-structures

G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein of positive scalar curvature and are uniquely determined by the simple symmetry group. During the proof the equations for G2-symplectic and G2-cosymplectic structures are studied and the topological types of the manifolds admitting such structures are determined. New examples of compact G2-cosymplectic manifolds and complete G2-symplectic structures are found.Comment: 23 page

On the geometry of closed G2-structure

We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Goldberg conjecture in almost Kahler geometry. The result was generalized by R.L.Bryant to closed G2-structures with too tightly pinched Ricci tensor. We extend it in another direction proving that a compact G2-manifold with closed fundamental form and divergence-free Weyl tensor is a G2-manifold with parallel fundamental form. We introduce a second symmetric Ricci-type tensor and show that Einstein conditions applied to the two Ricci tensors on a closed G2-structure again imply that the induced metric has holonomy group contained in G2.Comment: 14 pages, the Einstein condition in the assumptions of the Main theorem is generalized to the assumption that the Weyl tensor is divergence-free, clarity improved, typos correcte

The synergistic effects of saxagliptin and metformin on CD34+ endothelial progenitor cells in early type 2 diabetes patients: a randomized clinical trial.

AIMS: Type 2 diabetes is associated with endothelial dysfunction leading to cardiovascular disease. CD34+ endothelial Progenitor Cells (EPCs) are responsible for endothelial repair and neo-angiogenesis and can be used as a cardiovascular disease risk biomarker. This study investigated whether the addition of saxagliptin, a DPP-IV inhibitor, to metformin, may reduce cardiovascular disease risk in addition to improving glycemic control in Type 2 diabetes patients. METHODS: In 12 week, double-blind, randomized placebo-controlled trial, 42 subjects already taking metformin 1-2 grams/day were randomized to placebo or saxagliptin 5 mg. Subjects aged 40-70 years with diabetes for \u3c 10 years, with no known cardiovascular disease, BMI 25-39.9, HbA1C 6-9% were included. We evaluated EPCs number, function, surface markers and gene expression, in addition to arterial stiffness, blood biochemistries, resting energy expenditure, and body composition parameters. A mixed model regression to examine saxagliptin vs placebo, accounting for within-subject autocorrelation, was done with SAS (p \u3c 0.05). RESULTS: Although there was no significant increase in CD34+ cell number, CD31+ cells percentage increased. Saxagliptin increased migration (in response to SDF1α) with a trend of higher colony formation count. MNCs cytometry showed higher percentage of CXCR4 double positivity for both CD34 and CD31 positive cells, indicating a functional improvement. Gene expression analysis showed an upregulation in CD34+ cells for antioxidant SOD1 (p \u3c 0.05) and a downregulation in CD34- cells for IL-6 (p \u3c 0.01). For arterial stiffness, both augmentation index and systolic blood pressure measures went down in saxagliptin subjects (p \u3c 0.05). CONCLUSION: Saxagliptin, in combination with metformin, can help improve endothelial dysfunction in early diabetes before macrovascular complications appear. Trial registration Trial is registered under clinicaltrials.gov, NCT02024477