In this paper, we consider a Lie subalgebra p of the direct product gln×gln of the general linear Lie algebra. Let Sym(p) be the symmetric algebra equipped with the canonical linear Poisson bracket {⋅,⋅}0 induced by the Lie bracket on p. We define a new Liouville integrable system, i.e., a maximal Poisson commutative subalgebra, of the Poisson algebra (Sym(p),{⋅,⋅}0). We also construct the quantum integrable system, i.e., a maximal commutative subalgebra of the universal enveloping algebra U(p), corresponding to the new classical integrable system.
In this paper, we study anti-invariant Riemannian maps from Sasakian manifolds to Riemannian manifolds, with particular emphasis on the case where the structure vector field belongs to the horizontal distribution. We first establish a Chen–Ricci type inequality for such Riemannian maps. We then derive DDVV–type inequalities for anti-invariant Riemannian submersions from Sasakian space forms in terms of the vertical and horizontal distributions, respectively. Furthermore, we obtain a DDVV–type inequality for anti-invariant Riemannian maps from Sasakian manifolds to real space forms under the assumption that the structure vector field is horizontal. Finally, we construct non-trivial examples for anti-invariant Riemannian maps from Sasakian manifolds to Riemannian manifolds such that the structure vector field is a horizontal vector field. We also present examples that attain the equality cases of the Chen–Ricci type inequality for anti-invariant Riemannian maps and the DDVV–type inequality for anti-invariant Riemannian submersions.
Let M be a complete Riemannian manifold and V a smooth vector field on M. Under a lower bound on the k-Bakry–Émery Ricci curvature with k≥dimM, we derive an elliptic gradient estimate for positive bounded solutions of the fast diffusion equation associated with the V-Laplacianut=ΔVup,p∈(1−4k+4,1) on M×(−∞,+∞). As an application, a Liouville type theorem for positive ancient solutions is obtained. Our estimate differs from that in Tadano (2025) [22] and, when k≥5, it applies to a wider range of p, particularly on manifolds of dimension not less than five. Moreover, our Liouville type theorem partially improves the corresponding result in Jiang and Cheng (2019) [16].
In this paper, we introduce the notions of vertical and horizontal hypersubmersions with non-trivial examples, which are Riemannian submersions from complex space forms onto Riemannian manifolds such that the fibers and the horizontal spaces are the tangent bundles of real hypersurfaces of the source space, respectively. This research problem opens a new direction in the theory of Riemannian submersions and is essentially a dual analog of the concept of real hypersurfaces in complex space forms. First, treating the fibers of a Riemannian submersion as hypersurfaces, we determine the geometric properties of the fibers and the tensor field T. In this case, the Hopf hypersurface is also taken into account, and the notion of Hopf vertical hypersubmersion is introduced. Under this notion, we examine complex space forms, including complex projective spaces and complex hyperbolic spaces, in terms of their principal curvatures. Furthermore, the case in which the horizontal distribution of Riemannian submersions of co-dimension 1 is considered. In this case, we investigate the character of the tensor field A and the horizontal space, and examine the sectional curvatures of planes in the source and target spaces.
The well-known Li-Crampin theorem states that on weakly Berwald manifolds, a Finsler metric is a Landsberg metric if and only if it is a Berwald metric. In this paper, we extend this result to the class of (alpha, /3)-metrics. More precisely, we prove that an (alpha, /3)-metric with isotropic mean Berwald curvature is a generalized Landsberg metric if and only if it is a Berwald metric. As a consequence, every generalized Berwald C-reducible metric is a generalized Landsberg metric if and only if it is a Berwald metric. Furthermore, we show that any connected, positively (or negatively) complete generalized Berwald surface is a generalized Landsberg surface if and only if it is either Riemannian or locally Minkowskian, generalizing Vincze's result for Landsberg surfaces. Additionally, we construct a special pair consisting of a Riemannian metric and a 1-form, which can be used to generate infinitely many pure generalized Berwald metrics. We then investigate certain Riemannian and non-Riemannian curvature properties of the Randers metric derived from this pair. Finally, by providing an illustrative example, we demonstrate that the 2-dimensional Bartelme ss-Lang rigidity theorem does not extend to higher dimensions. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We prove local existence of solutions in the case of cosmological models for the Einstein-Vlasov-scalar field system with two-torus symmetry in expanding direction. The sources of equations are generated by a distribution function and scalar field, subject to Vlasov and nonlinear wave equations respectively. We use a priori estimations and iterations on short-time existence theorems for the partial differential equations resulting from the complete system in areal coordinates. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study an overdetermined problem for the first nonzero p-Steklov eigenvalue on a domain in a Riemannian manifold with non-negative Ricci curvature, which is an extension of Lee-Seo's result [18]. In particular, we characterize a geodesic ball in R n via the first non-zero p-Steklov eigenfunction satisfying an overdetermined problem.
We extend a class of the Kazdan-Warner equations to the network Gamma with the Kirchhoff condition and discuss the existence of their solutions. Firstly, we extend the Delta u = K-fe(2u) to the network, where K(x) is a continuous function. Even though this equation can be converted by some methods to the case where K(x) is constant, we will also give a direct proof with the cases integral(Gamma) K > 0, integral(Gamma) K = 0, integral(Gamma) K < 0. Moreover, we also extend another type of the Kazdan-Warner equation Delta u = alpha u-fe(2u) and give some results. Particularly, when alpha > 0 and f <= 0, our conclusion is complete. Finally, we extend the following equations: Delta u = beta he(u) /integral(m) he(u)d mu + beta/Vol(M) and Delta u + alpha u = - beta he(u)/integral(M) he(u)d mu + beta/ Vol(M) to the network. We will discuss the existence of solutions to the above two equations by considering the functional J(alpha,beta) with the cases alpha> -lambda(k), alpha = -lambda(k) and alpha < -lambda(k), where lambda(k) is the k-th nonzero eigenvalue of L = -partial derivative(2). Particularly, the case alpha = -lambda(k) is slightly more complicated. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We prove the rigidity results for G-equivariant harmonic maps of the sphere Sm into real or complex Grassmannians of low rank for G = SO(m + 1), Spin(m + 1), G2 (when m = 6) and Spin(7) (when m = 7). (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
This article explores Einstein-type structures on contact metric manifolds. We first establish that a closed Einstein-type structure on a complete K-contact or Sasakian manifold is a compact Einstein manifold. Subsequently, several adequate criteria are established to determine when a complete K-contact manifold with an Einstein-type structure is trivial (eta-Einstein). Following that, certain results for H-contact and complete contact manifolds are illustrated. Next, we establish that a non-Sasakian (k, mu)-contact manifold possessing a non-degenerate closed Einstein-type structure is flat in three dimensions and is locally isometric to Rn+1 x S-n(4) for higher dimensions. Finally, we provide a couple of applications of Einstein-type solitons in general relativity. This includes characterizing perfect fluid space-time equipped with an Einstein-type structure as dark energy era or having vanishing vorticity. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In the spirit of Witten's gauging of exterior differential, we introduce deformed Lie derivative and differential operator by using a-symmetries. We will provide a new geometric property of a-symmetries by using deformed Lie derivative and show that a-symmetries can yield closed 1-forms in the sense of deformed differential operator. In addition, by using the deformation Lie derivative, we give the equivalence condition under which the set of vector fields is a a-symmetry and also we give several theorems to find first integrals of the system considered. Especially, we give a first integral for some ordinary differential equation by these theorems. Moreover, we use deformed differential operator to get some complexes, and we study the cohomology groups of them. Further, we get that equivalence condition that a closed 1-form is closed in the sense of deformed differential operator if there is a a-symmetry for the system under consideration. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We develop a complete Weierstrass-Kenmotsu type representation for conformal immersions of constant mean curvature 0 <= H < 1 in the hyperbolic 3space H-3(-1). Our construction rests on three geometric pillars: the Hermitian model of H-3(-1), the global Iwasawa splitting SL(2, C) = S & centerdot; SU(2), and Kokubu's adjusted Gauss map into the 2-sphere with its Kobayashi-type metric. We prove that every such immersion is locally obtained from a rank-one (1,0)-form eta via a flat SL(2, C)-connection S(-1)dS = eta - lambda eta(& lowast;) with constant lambda is an element of C-& lowast;, and that the mean curvature is H = 1-|lambda|(2)/1+|lambda|(2) . We give an explicit connection to the Aiyama-Akutagawa formula, analyze period or monodromy issues and completeness of ends, and provide model families together with a practical DPW-style algorithm (balanced gauge, Iwasawa factorization, unitarization). A fully worked two-ended example with closing conditions, a stability (second variation) analysis for symmetric annuli, and rigorous numerical error estimates are included. Appendices supply balancing derivations, symbolic exponentials, auxiliary proofs, and a Jacobi-operator derivation in Aiyama-Akutagawa variables. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let (M, F, m) be a forward complete and noncompact Finsler measure space with weighted Ricci curvature bounded from below. In this paper, we give local gradient estimates of positive weak solutions for a class of nonlinear p(> 1)-Laplacian equations involving parameters p, a, sigma on (M, F, m). As applications, we obtain the nonexistence of positive weak solutions for this class of equations under some ranges of p, a, sigma and Harnack inequalities. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let (M1, F1) and (M2, F2) be a pair of Finsler manifolds. The Minkowskian product Finsler manifold (M, F) of (M1, F1) and (M2, F2) with respect to a function f is the product manifold M = M1 & times; M2 endowed with the Finsler root metric F = f (S, H), where S = F12, H = F 2 2 . In this paper, we prove that a Minkowskian product Finsler manifold has isotropic flag curvature if and only if it has vanishing Riemann curvature, and prove that a Minkowskian product Finsler manifold (M, F) is Ricci-flat if it has scalar flag curvature. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we study Finsler gradient almost Ricci solitons of spherically symmetric type. We find the PDE characterization for a spherically symmetric Finsler manifold with spherically symmetric volume form to be a Finsler gradient almost Ricci soliton. As an application, we classify all polynomial spherically symmetric Finsler gradient almost Ricci solitons. These solitons must be steady or Douglas. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
For a complete open Riemannian manifold with nonnegative Ricci curvature, we show that the dimension of the space of ancient biharmonic caloric functions with polynomial growth is bounded by the degree of growth times the dimension of biharmonic functions with the same growth. Moreover, we investigate biharmonic caloric functions on Euclidean spaces. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
