Narrow Results

In this work we show that for the geodesic spray S of a Finsler function F, the most natural projective deformation leads to a non-Finsler metrizable spray, for almost every value of λ ∈ ℝ. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence, we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn–Minkowski inequality. A reverse isoperimetric inequality for convex bodies is also given.

A discussion is given of the sectional curvature function on a four-dimensional Lorentz manifold and, in particular, on the space–time of Einstein's general relativity theory. Its tight relationship to the metric tensor is demonstrated and some of its geometrical and algebraic properties evaluated. The concept of a sectional curvature preserving symmetry, in the form of a certain smooth vector field, is introduced and discussed.

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.

We introduce a study of Riemannian manifold M = ℝ² endowed with a metric of diagonal type of the form , where g is a positive function, of C^∞-class, depending on the variable x² only. We emphasize the role of metric in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

This paper aims to be a natural continuation of our very recent research, published in [Int. J. Geom. Meth. Mod. Phys.8(4) (2011) 783–796]. Using as geometric background a generalized Poincaré manifold, this work presents models of new explicit classes of manifolds within the class of gradient Ricci solitons. In Sec. 1 we shortly introduce a history of this topic to motivate its relevance both for theoretical and applied sciences, and in Sec. 2 our work contains its notations and assumptions. In Sec. 3 we give our framework and introduce the basic result, and in Sec. 4 we develop our theory. Finally, in Sec. 5, we conclude this paper and suggest possible further development. While the background in the first two sections is introductory, the theory in Secs. 3 and 4 is new as a whole, containing original results.

On a pseudo-Riemannian manifold (M, g) we consider ∇ the Levi-Cività connection associated to metric g and a function f : M → ℝ whose pseudo-Riemannian Hessian is non-degenerate and with constant signature. We study properties of the pseudo-Riemannian manifold (M, h), in terms of local computation. Investigating the conditions for existence of the equal connections , produced by g and h, we determine classes of explicit Ricci flat gradient solitons for some particular forms of g, arising from remarkable physics models.

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp’s theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.

Let $(T_1{𝕄}_1^2,g_1)$ be a unit tangent bundle of Minkowski surface $({𝕄}_1^2,g)$ endowed with the pseudo-Riemannian induced Sasaki metric. In this present paper, we studied the N-Legendre and N-slant curves in which the inner product of its normal vector and Reeb vector is zero and nonzero constant, respectively, in $\left(\right.T_1{𝕄}_1^2,g_1\left)\right.$ and several important characterizations of these curves are given.

The definition of Schrödinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrödinger flow of maps into S², and that there exists a unique local smooth solution for the initial value problem of one-dimensional Schrödinger flow of maps into Kähler manifolds. In case the targets are Kähler manifolds with constant curvature, it is proved that one-dimensional Schrödinger flow admits a unique global smooth solution.