We investigate Möbius isoparametric hypersurfaces in the (n+1)-Euclidean unit sphere ‫ޓ‬ n+1 with three distinct Möbius principal curvatures. As direct consequence of our main result, we establish the complete classification for all such hypersurfaces in ‫ޓ‬ 6 .

Background Gut microbiota plays an important role in the development of atopic dermatitis (AD). We aimed to elucidate research trends in gut microbiota and AD in children, to provide evidence and insights to the clinical prevention and treatment of AD in children. Methods A scoping literature review on the studies of gut microbiota and AD were conducted. Two authors independently searched Pubmed et al. databases for studies focused on gut microbiota and AD in children up to January 15, 2022. The literatures were screened and analyzed by two reviewers. Results A total of 44 reports were finally included and analyzed. Current researches have indicated that abnormal human microecology is closely associated with AD, and the disturbance of intestinal microbiota plays an important role in the occurrence and development of AD. Probiotics can correct the microbiota disorder, have the functions of regulating immunity, antioxidant, and help to restore the microecological homeostasis. However, there is still a lack of high-quality research reports on the efficacy and safety of probiotics in the prevention and treatment of AD in children. Conclusions The changes of gut microbiota are essential to the development of AD in children, which may be an effective target for the prevention and treatment of AD. Future studies with larger sample size and rigorous design are needed to elucidate the effects and safety of probiotics in AD.

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An immersed umbilic-free hypersurface in the unit sphere is equipped with three Möbius invariants, namely, the Möbius metric, the Möbius second fundamental form and the Möbius form. The fundamental theorem of Möbius submanifolds geometry states that a hypersurface of dimension not less than three is uniquely determined by the Möbius metric and the Möbius second fundamental form. A Möbius isoparametric hypersurface is defined by two conditions that it has vanishing Möbius form and has constant Möbius principal curvatures. It is well-known that all Euclidean isoparametric hypersurfaces are Möbius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, combining with previous results, a complete classification for all Möbius isoparametric hypersurfaces in the unit six-sphere is established.

In this paper, we study umbilic-free submanifolds of the unit sphere with parallel Möbius second fundamental form. As one of our main results, we establish a complete classification for such submanifolds under the additional condition of codimension two. Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by MICHIGAN STATE UNIVERSITY on 02/07/15. For personal use only. 1450062-2 Int. J. Math. 2014.25. Downloaded from www.worldscientific.com by MICHIGAN STATE UNIVERSITY on 02/07/15. For personal use only. Submanifolds with parallel Möbius second fundamental form (d) the image of σ of a cone overwhere H n−k ( √ 1 + a 2 ) denotes the (n − k)-dimensional hyperbolic space of constant sectional curvature −1/(1 + a 2 ), and S k (a) denotes the k-dimensional Euclidean sphere of radius a.Remark 1.1. The two maps σ and τ mentioned above are the canonical conformal diffeomorphisms whose definitions are presented in (2.2) and (2.3), respectively. Whereas the notion about "cone" is referred to Definition 3.1.When compared with the hypersurfaces situation, the problem on the classification of higher codimension umbilic-free submanifolds of the unit sphere with parallel Möbius second fundamental form is now still open, which obviously would be much more complicated.In this paper, as our continuous attempts working on the above problem, we made the discovery that the assumption "parallel Möbius second fundamental form" implies that the submanifold is of parallel Blaschke tensor. This latter property plays an important role for us to know further about submanifolds with parallel Möbius second fundamental form. Indeed, as our main result, on the basis of Ferus, Backes-Reckziegel and Takeuchi's classification of (Euclidean) parallel submanifolds in space forms, and by using the particular properties of submanifolds of codimension two, a complete classification of Möbius parallel submanifolds can be established for p = 2 case, which can be stated as follows:Theorem 1.2. Let x : M n → S n+2 be an n-dimensional umbilic-free submanifold with parallel Möbius second fundamental form. Then x is Möbius equivalent to an open part of one of the following submanifolds:(i) an umbilic-free submanifold with parallel second fundamental form in S n+2 , (ii) the image of σ of an umbilic-free submanifold with parallel second fundamental form in the Euclidean space R n+2 , (iii) the image of τ of an umbilic-free submanifold with parallel second fundamental form in the hyperbolic space H n+2 , (iv) the image of σ of a cone over some k-dimensional umbilic-free submanifold with parallel second fundamental form in S k+2 (k ≤ n − 1).Remark 1.2. In above theorem, we made the assumption of codimension two that is used, first of all, in Lemma 5.1 to reduce the distinct number of eigenvalues of A to no more than four. We noticed, which Prof. Xiang Ma also observed and kindly informed us, that in higher codimensional case similar results can still be obtained, but the problem at the moment is that subsequently geometric discussions become much more ...