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This paper is concerned with the study of n-dimensional conformally flat Riemannian manifolds for n ≥ 3 and its applications in affine differential geometry. First, improving the work of Sekigawa–Takagi (Tohoku Math J 23:1–11, 1971), we have a complete classification for conformally flat Riemannian manifolds with semi-parallel Ricci tensor. Then, as an application, we establish a complete classification of locally strongly convex affine hyperspheres in the (n + 1) -dimensional affine space R n + 1 with conformally flat affine metric and semi-parallel Ricci tensor, which generalizes the previous works of Cheng–Hu–Moruz–Vrancken (Sci China Math 63:2055–2078, 2020) and Hu–Xing (J Math Anal Appl 528:127596, 2023) on affine hyperspheres with parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

In this paper, we study real hypersurfaces of the two Kähler surfaces S 2 × S 2 and H 2 × H 2. As the main results, amongst others, we classify all such hypersurfaces whose normal Jacobi operators are parallel with respect to either the Levi-Civita connection or the k-generalized Tanaka–Webster connection. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of n-dimensional Möbius semi-parallel submanifolds in the (n + p) -dimensional unit sphere. As the results, we first show that for such submanifolds the covariant derivatives of their Möbius form are symmetric, their Blaschke tensor are also semi-parallel, and the number of distinct Blaschke eigenvalues is at most (p + 2) . Then, extending well known results we establish an optimal rigidity theorem characterizing several typical examples of such submanifolds. Finally, when restricting to hypersurfaces, i.e., p = 1 , we first show that the number of distinct Möbius principal curvatures is at most 3, and then we present the classification theorem for Möbius semi-parallel hypersurfaces with two distinct Möbius principal curvatures. We also study Möbius semi-parallel hypersurfaces with three distinct Möbius principal curvatures, and partial classification results are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we study 2-Hopf hypersurfaces in nonflat complex planes M ¯ 2 (c) . First, we prove that an η -parallel 2-Hopf hypersurface in nonflat complex planes is a ruled hypersurface. Second, we prove that a hypersurface M in M ¯ 2 (c) is a strongly 2-Hopf ruled hypersurface if and only if M is an η -parallel 2-Hopf hypersurface with the function α : = g (A ξ , ξ) being a constant along the structure vector field ξ . Finally, we prove that there do not exist locally conformally flat 2-Hopf hypersurfaces in nonflat complex planes. [ABSTRACT FROM AUTHOR]

In [Int Math Res Not 7: 5492–5519 (2021)], Feng and He introduced the Orlicz Aleksandrov problem and proved that, there exists a convex body and an explicit constant, such that the problem is solved provided the given measure is even. In this paper, extending their results, we first prove that the desired convex body always exists for each constant in an interval. Then, we prove the existence of solutions to the problem for general measure that is not necessarily even. [ABSTRACT FROM AUTHOR]

In [J Diff Equ, 269: 8549–8572, 2020], Li and Zhu studied the Orlicz–Minkowski problem for torsional rigidity, and among other things, they proved the existence of solutions to the problem regarding a continuous function φ satisfying lim x → 0 + φ (x) = ∞ . In this paper, with the motivation of complementing their results, we prove a new existence of solutions to the problem regarding a strictly increasing, continuously differentiable function φ satisfying lim x → 0 + φ (x) = 0 . [ABSTRACT FROM AUTHOR]

Substorm processes have been studied in detail, and it is well known that interplanetary (IP) shock encountering the terrestrial magnetosphere causes global responses. However, how IP shock compression to the magnetosphere affects the development of an ongoing substorm remains uninvestigated. Herein, the simultaneous satellite and ground-based auroral evolutions associated with an IP shock impact on the magnetopause during an ongoing substorm on May 7th, 2005, were examined. The IMAGE satellite over the Southern Hemisphere captured the global development substorm, which was initiated at 17:38:47 UT. The poleward branch of the nightside auroral oval was fortuitously monitored by an all-sky camera at the Zhongshan Station (−74.5° magnetic latitude, ZHO) in Antarctica. The satellite imager observed continuous brightening and broadening of the nightside auroral oval after the IP shock arrival. The simultaneous ground-based optical aurora measurement displayed the intensification and expansion of a preexisting auroral surge poleward of the aurora oval. The geomagnetic field variations and the instantly increased PC indices indicated an elevated merging rate and enhanced the convection-related DP-2 currents. Therefore, this IP shock transient impact did not significantly change the ongoing development of the substorm, although it meets the magnetospheric precondition hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we study the interesting open problem of classifying the locally strongly convex centroaffine Tchebychev hypersurfaces in R n + 1 with constant sectional curvature. First, for arbitrary dimensions we solve the problem by assuming that the centroaffine shape operator vanishes. Second, extending the solved cases of n = 2 , 3 , we continue working with the case n = 4 . As the result, we establish a complete classification of the flat hyperbolic centroaffine Tchebychev hypersurfaces in R 5 . [ABSTRACT FROM AUTHOR]

In this paper, improving the results in [Ann Mat Pura Appl 199 (2020), 1147–1170] and [Manuscr Math, https://doi.org/10.1007/s00229-021-01308-4], we classify hypersurfaces of the homogeneous nearly Kähler manifold S 3 × S 3 of which holomorphic distributions are preserved by the canonical almost product structure P of S 3 × S 3 . We also characterize these hypersurfaces as Hopf ones with constant Reeb functions and hypersurfaces with weakly η -parallel shape operators. [ABSTRACT FROM AUTHOR]

In this sequel to our article on Hopf hypersurfaces of the homogeneous NK (nearly Kähler) manifold S 3 × S 3 (Hu and Yao in Ann Mat Pura Appl, 199:1147–1170, 2020), we continue working on the problem of determining the Hopf hypersurfaces of this ambient space for which the holomorphic distributions are preserved by the almost product structure of S 3 × S 3 . As complements to existing results, we first show that no such Hopf hypersurfaces admit exactly four distinct principal curvatures. Then, as the second main result and an important step towards the solution of the preceding problem, we establish a complete classification of such Hopf hypersurfaces with five distinct constant principal curvatures. [ABSTRACT FROM AUTHOR]

In this paper, based on the classical Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction J H and the norm of the covariant differentiation of the second fundamental form h, where J is the almost complex structure and H is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure (φ , ξ , η , g) , we also establish an optimal integral inequality involving the Ricci curvature in the direction φ H and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant. [ABSTRACT FROM AUTHOR]

Lee and Suh (Proc R Soc Edinb Sect A, 2021, ) proved that there do not exist Hopf hypersurfaces with parallel Ricci tensor in the complex quadric Q m = S O m + 2 / S O m S O 2 for m ≥ 4 . In this paper, extending the result of Lee and Suh, we show that, for m ≥ 3 , Q m does not admit Hopf hypersurfaces with recurrent Ricci tensor. This implies that there do not exist Hopf hypersurfaces in Q m (m ≥ 3) with parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

In this paper, sharpening a theorem of Yamaguchi et al (J Diff Geom 11: 59–64, 1976), we give a complete classification of n-dimensional C-totally real minimal submanifolds with constant sectional curvature in the (2 n + 1) -dimensional Sasakian space forms. [ABSTRACT FROM AUTHOR]

In this paper we consider minimal Lagrangian submanifolds in n-dimensional complex space forms. More precisely, we study such submanifolds which, endowed with the induced metrics, write as a Riemannian product of two Riemannian manifolds, each having constant sectional curvature. As the main result, we give a complete classification of these submanifolds. [ABSTRACT FROM AUTHOR]

In this short note, studying 3-dimensional compact and minimal submanifolds of the (3 + p) -dimensional unit sphere S 3 + p (1) , we establish two rigidity theorems in terms of the Ricci curvature. The first theorem related to hypersurfaces of S 4 (1) gives a new characterization of the minimal Clifford torus, whereas the second theorem is about the Legendrian submanifolds of S 7 (1) so that a new characterization of the Calabi torus can be presented. [ABSTRACT FROM AUTHOR]

Let N 2 n + 1 (c) be an (2 n + 1) -dimensional Sasakian space form with Sasakian structure (φ , ξ , η , g) and constant φ -sectional curvature c. An n-dimensional submanifold M n of N 2 n + 1 (c) is called integral if the contact form η restricted to M n vanishes. In this paper, we established a general inequality for n-dimensional integral submanifolds in Sasakian space forms involving the norm of the covariant differentiation of both the second fundamental form h and the mean curvature vector field H. Our result is optimal in that we can classify all integral submanifolds realizing the equality case of the inequality. As direct consequence, we give a characterization of the contact Whitney spheres in the Sasakian space forms. [ABSTRACT FROM AUTHOR]

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ℝn+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3- and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified. [ABSTRACT FROM AUTHOR]

In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343–373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly Kähler) manifold S 3 × S 3 . First, we show that any Hopf hypersurface of the homogeneous NK S 3 × S 3 does not admit two distinct principal curvatures. Then, for the important class of Hopf hypersurfaces with three distinct principal curvatures, we establish a complete classification under the additional condition that their holomorphic distributions { U } ⊥ are preserved by the almost product structure P of the homogeneous NK S 3 × S 3 . [ABSTRACT FROM AUTHOR]

In this short paper, we study compact Lagrangian submanifolds of the homogeneous nearly Kähler 6-dimensional unit sphere S 6 (1) . Following a strategy in Hu et al. (J Geom Phys 144:199–208, 2019), we shall establish an optimal pinching theorem in terms of the Ricci curvature so that a new characterization of the totally geodesic S 3 (1) and the Dillen–Verstraelen–Vrancken's Berger sphere S 3 (described in J Math Soc Jpn 42:565–584, 1990) in S 6 (1) can be given. [ABSTRACT FROM AUTHOR]

The equivariant CR minimal immersions from the round 3-sphere S3 into the complex projective space CPn have been classified by the third author explicitly (Li in J Lond Math Soc 68:223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant and that all geometric properties of S3=SU(2) endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra su(2), we establish an extended classification theorem for equivariant CR minimal immersions from the 3-sphere S3 into CPn without the assumption of constant sectional curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $$\mathbb {R}^{n+1}$$ involving the norm of the covariant differentiation of both the difference tensor K and the Tchebychev vector field T. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in Cheng et al. (Results Math, doi:, 2017), we can completely classify the hypersurfaces which realize the equality case of the inequality. [ABSTRACT FROM AUTHOR]

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu et al. (J Differ Geom 87:239-307, 2011). [ABSTRACT FROM AUTHOR]

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S × S is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((∇ h)( v, v, v), Jv) = λ holds for all unit tangent vector v. [ABSTRACT FROM AUTHOR]

A new technique for estimating quiet day curve (QDC) was suggested. To validate the new approach, QDCs were derived from data acquired by the imaging riometer installed at the Chinese Zhongshan Station in Antarctic. The evaluation was performed by comparing the difference between QDC derived by the new technique and those derived by Tanaka's technique. The results were discussed in terms of the diurnal variation and discrepancy. Also, cosmic noise absorption (CNA) images were built using both techniques in order to evaluate the implications of the changes when obtaining them using different methods of the QDC determination. The influence of the multiplicative factor value on the QDC determination in Tanaka's technique was evaluated. [ABSTRACT FROM AUTHOR]

Using high temporal resolution optical data obtained from three-wavelength all-sky imagers over six winters continuously at Yellow River Station (78.92°N, 11.93°E) in Arctic, we statistically investigated the dependence of location of poleward moving auroral forms (PMAFs) on the interplanetary magnetic field (IMF) B and B components as a function of MLT and MLAT under stable IMF conditions. It is found that more PMAFs occurred in lower latitude for B < 0 and there was less evident IMF B-related prenoon-postnoon asymmetry for B < 0 than for B > 0. We found that the PMAFs were distributed over a wide range of MLT when B < 0, which indicates that the reconnection X-line might spread like an 'S'' shape. However, during northward IMF, PMAFs were observed predominantly prenoon for IMF B > 0 and postnoon for IMF B < 0 associating with the effect of the high-latitude reconnection, which is largely consistent with the theoretical model of the convection flow. [ABSTRACT FROM AUTHOR]

An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Möbius form vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we give a complete classification for all Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues. [ABSTRACT FROM AUTHOR]

We study Lorentzian affine hypersurfaces of $${\mathbb{R}^{n+1}}$$ having parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric. As main result, we obtain a complete classification of these hypersurfaces. [ABSTRACT FROM AUTHOR]

Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics. [ABSTRACT FROM AUTHOR]

We show that a Riemannian metric on a 4-dimensional smooth manifold is of constant scalar curvature if and only if it is a critical metric of the restricted Schouten functional. [ABSTRACT FROM AUTHOR]

There exists a well known construction which allows to associate with two hyperbolic affine hyperspheres $$f_{i} : M^{n_{i}}_{i} \rightarrow {\mathbb{R}}^{n_{i}+1}$$ a new hyperbolic affine hypersphere immersion of $$I \times M_{1} \times M_{2}$$ into $${\mathbb{R}}^{n_{1}+n_{2}+2}$$ . In this paper we deal with the inverse problem: how to determine from properties of the difference tensor whether a given hyperbolic affine hypersphere immersion of a manifold $$M^{n} \rightarrow R^{n+1}$$ can be decomposed in such a way. [ABSTRACT FROM AUTHOR]

Let M n( n ≥ 2) be an immersed umbilic-free hypersurface in the ( n+1)-dimensional unit sphere S n+1. Then M n is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M nunder the Möbius transformation group of S n+1. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form. [ABSTRACT FROM AUTHOR]