arXiv subjects
Publications and source records attributed to Dan Gregorian Fodor.
Let $n \geq 3$ and $R_{abcd}$ be a $(4,0)$ sectionally positive curvature-type tensor (a tensor possessing all the local symmetries of the $(4,0)$ curvature tensor). Then there exists a metric tensor $g_{ab}$ such that $R_{abcd}\; g^{bd} = g_{ac} λ$ for some $λ$. Furthermore, $g_{ab}$ is unique up to a constant factor.
In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators in Finsler geometry, [4,7]. This algebraic characterisation for Finsler metrics of constant flag curvature allows to provide yet another proof for the Finslerian version of Beltrami's Theorem, [2,3].
We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields on the manifold, satisfying a certain non-linear equation involving the Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the fundamental theorem of hypersurfaces. In the case of manifolds of positive sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and Codazzi equations to the cancelation of a set of obstructions involving the logarithm of the Riemann curvature operator. The resulting theorem has a structural similarity to the Weyl-Schouten theorem, suggesting a parallelism between conformally flat $n$-manifolds and those that admit an isometric immersion in $\mathbb{R}^{n+1}$.
We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension $n=4$, and complete characterization for a dense open subset of the space of operators in dimension $4$. We also briefly examine higher-dimentional curvature operators.
In this note some properties of the sum of element orders of a finite abelian group are studied.
Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We give expressions of $m$'s associated connection, and Riemann curvature tensor $R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of $m$ (with respect to the Levi-Civita connection associated with $g$). The formulas turn out to be generalizations of the coordinate expressions. Coordinate expression formulas can be recovered from ours by setting $g$ as the Euclidean metric induced by a given coordinate chart. As the covariant derivative induced by $g$ becomes the ordinary partial derivative and the $R_g$ tensor vanishes, the formulas coincide with the well-known coordinate expressions for $m$'s connection and curvature tensor.
The main goal of the current paper is to determine the total number of diamonds in the subgroup lattice of a finite abelian group. This counting problem is reduced to finite $p$-groups. Explicit formulas are obtained in some particular cases.