In this paper, we prove two generalized versions of the Cheeger-Gromoll splitting theorem via the non-negativity of the Bakry-Émery Ricci curavture on complete Riemannian manifolds.

The vascular endothelial glycocalyx is a dense, bush-like structure that is synthesized and secreted by endothelial cells and evenly distributed on the surface of vascular endothelial cells. The blood-brain barrier (BBB) is mainly composed of pericytes endothelial cells, glycocalyx, basement membranes, and astrocytes. The glycocalyx in the BBB plays an indispensable role in many important physiological functions, including vascular permeability, inflammation, blood coagulation, and the synthesis of nitric oxide. Damage to the fragile glycocalyx can lead to increased permeability of the BBB, tissue edema, glial cell activation, up-regulation of inflammatory chemokines expression, and ultimately brain tissue damage, leading to increased mortality. This article reviews the important role that glycocalyx plays in the physiological function of the BBB. The review may provide some basis for the research direction of neurological diseases and a theoretical basis for the diagnosis and treatment of neurological diseases.

A quaternionic Kähler manifold M is called positive if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a positive quaternionic Kähler manifold. As applications we prove that, among others, a 4m-dimensional positive quaternionic Kähler manifold with symmetry rank at least (m − 2) must be either isometric to HP m or Gr 2 (C m+2 ), if m ≥ 10. IntroductionA quaternionic Kähler manifold M is an oriented Riemannian 4n-manifold, n ≥ 2, whose holonomy group is contained in Sp(n)Sp(1) ⊂ SO(4n). If n = 1 we add the condition that M is Einstein and self dual. Equivalently, there exists a 3-dimensional subbundle S, of the endmorphism bundle End(T M, T M ) locally generated by three anti-commuting almost complex structures I, J, K = IJ so that the Levi-Civita connection preserves S. It is well-known [Ber] that a quaternionic Kähler manifold M is always Einstein, and is necessarily locally hyperKähler if its Ricci tensor vanishes. Motivated by the Penrose twistor construction (cf. [AHS] [Hi]), Salamon [Sa] developped the important twistor space theory for quaternionic Kähler manifolds, showing that the unit sphere bundle of S, called the twistor space Z, admits a complex structure so that the fiber of the P 1 -fibration p : Z → M is a rational curve. A quaternionic Kähler manifold M is called positive if it has positive scalar curvature. By [Hi] (for n = 1) and [Sa] (for n ≥ 2, compare [Le] [Le-Sa]) a positive quaternionic Kähler manifold M has twistor space Z a complex Fano manifold. Hitchin [Hi] proved a positive quaternionic Kähler 4-manifold M must be isometric to CP 2 or S 4 . Hitchin's work was extended by Poon-Salamon [PS] 1 Supported by NSF Grant 19925104 of China, 973 project of Foundation Science of China, RFDP Typeset by A M S-T E X to dimension 8, showing that a positive quaternionic Kähler 8-manifold M must be isometric to HP 2 , Gr 2 (C 4 ) or G 2 /SO(4). This leads to the Salamon-Lebrun conjecture: Every positive quaternionic Kähler manifold is a quaternionic symmetric space. Very recently, the conjecture was further verified for n = 3 in [HH], using approach initiated in [Sa] [PS] (compare [LeSa]). For a positive quaternionic Kähler manifold M , Salamon [Sa] proved that the dimension of its isometry group is equal to the index of certain twisted Dirac operator, which by the Atiyah-Singer index theorem, is a characteristic number of M coupled with the Kraines 4-form Ω (in analog with the Kähler form), and it was applied to prove the isometry group of M is large in lower dimensions (up to dimension 16). By [LeSa] any positive quaternionic Kähler 4n-manifold M is simply connected and the second homotopy group π 2 (M ) is a finite group or Z, and M is isometric to HP n or Gr 2 (C n+2 ) according to π 2 (M ) = 0 or Z. The main purpose of this paper is to prove several connectedness theorems for positive quaterni...

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A long-standing conjecture of Farrell and Zdravkovska and independently S. T. Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions.