International Journal of Mathematics

Every Riemannian metric or Finsler metric on a manifold induces a spray via its geodesics. In this paper, we discuss several expressions for the $\chi$-curvature of a spray. We show that the sprays obtained by a projective deformation using the S-curvature always have vanishing $\chi$-curvature. Then we establish the Beltrami Theorem for sprays with $\chi =0$.

In the proof of Theorem 3.7 of the original paper Internat. J. Math. 30(14) (2019) 1950078, 1–25, two inequalities are used that do not hold in general. In this note, we prove an extra propositions which allows us to give a proof of Theorem 3.7 without using the aforementioned inequalities. Hence, all the results in the original paper are valid. We have also posted a corrected version in the arXiv.

The purpose of this work is to study the following singular problem:

In this work, we study the fractional Laplacian equation with singular nonlinearity:

For a homogeneous space $G/H$ of reductive type, we consider the tangential homogeneous space $G_{𝜃}/H_{𝜃}$. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ($=$obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact $4$-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: $𝔯{𝔯}_{3,-1}$, ${𝔫𝔦𝔩}^4$ and ${𝔯}_{4,\lambda ,-(1+\lambda )}$ with $-1<\lambda <-\frac{1}{2}$. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover, we prove Ricci flatness of the canonical connection for the almost Kähler structure. Regarding the other two manifolds we compute the Kodaira dimension for certain almost complex structures. Finally, we construct a natural hypercomplex structure providing a twistorial description.

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray $S$ on a Lie group $G$ satisfies that $G$ is commutative or $S$ is projective, then $S$ is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that ${\bigcup }_{k\ge 1}{M}_k(D)$ admits a factor trace if and only if $D\in \{2cos(\pi /n)+1|n\ge 3\}\cup [3,\infty )$ and the higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over these factors is constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.

We establish the twisted crystallographic T-duality, which is an isomorphism between Freed–Moore twisted equivariant K-groups of the position and momentum tori associated to an extension of a crystallographic group. The proof is given by identifying the map with the Dirac homomorphism in twisted Chabert–Echterhoff KK-theory. We also illustrate how to exploit it in K-theory computations.