The pseudo-Riemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Sta... more The pseudo-Riemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Statistics as an important tool for recent research. Given a pseudo-Riemannian manifold (M, g) and a smooth function f : M → R, whose Hessian with respect to g is non-degenerate, one can define on M the associated pseudo-Riemannian Hessian metric h = Hess g f . In the following, we apply this methodology for describing the geometrical properties of some interesting mathematical objects like the higher dimensional Reissner-Nördstrom black holes. The paper is organized as follows. Section 1 reviews the history of pseudo-Riemannian Geometry and points out which Hessian is more convenient for physical problems. Section 2 gives the Christoffel symbols and the system of geodesics of pseudo-Riemannian manifold (M, h = Hess g f ), establishes the relation between the components of the curvature tensors field of (M, h) and (M, g), and determines the PDEs representing the coincidence between the Christoffel symbols of (M, h) and the Christoffel symbols of (M, g). The last section presents the comparison of null-length curves trajectories obtained with the two metrics for a 5-dimensional RN black hole.
ABSTRACT The guest editors would like to express their gratitude to the reviewers for their serio... more ABSTRACT The guest editors would like to express their gratitude to the reviewers for their serious evaluation of the papers submitted in this special issue. Thanks also go to the authors for their interesting papers. Carlo Bianca acknowledges the support by the FIRB project RBID08PP3J (Metodi matematici e relativi strumenti per la modellizzazione e la simulazione della formazione di tumori, competizione con il sistema immunitario, e conseguenti suggerimenti terapeutici).
With the end positions of the stroke and the radius of the crank specified, the locus of the fixe... more With the end positions of the stroke and the radius of the crank specified, the locus of the fixed end of the crank in the plane of crank rotation can be graphically or analytically determined.
Section 1 recalls some basic notions of the geometry of jet bundles. Section 2 analyses the conne... more Section 1 recalls some basic notions of the geometry of jet bundles. Section 2 analyses the connection between least squares problem for ODEs and the geometric dynamics. Section 3 discusses the method of variational integrators for the Lagrangian dynamics and adopts a simplified ...
This paper investigates the way in which geometric dynamics on Riemannian manifolds can be applie... more This paper investigates the way in which geometric dynamics on Riemannian manifolds can be applied to fluid flow. The equations of motion from fluid mechanics (momentum equations), expressed on general curvilinear coordinates, are compared with the equations of motion in a gyroscopic field of forces. Attempts are made to seek conditions under which both describe the same motion, especially in the case when trajectories are not field lines. It is proved that all pathlines and streamlines are geodesics in the least square sense and, in some circumstances, other trajectories included in geometric dynamics have physical support.
The pseudo-Riemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Sta... more The pseudo-Riemannian Geometry is utilized in Mathematical Optimization, Thermodynamics or in Statistics as an important tool for recent research. Given a pseudo-Riemannian manifold (M, g) and a smooth function f : M → R, whose Hessian with respect to g is non-degenerate, one can define on M the associated pseudo-Riemannian Hessian metric h = Hess g f . In the following, we apply this methodology for describing the geometrical properties of some interesting mathematical objects like the higher dimensional Reissner-Nördstrom black holes. The paper is organized as follows. Section 1 reviews the history of pseudo-Riemannian Geometry and points out which Hessian is more convenient for physical problems. Section 2 gives the Christoffel symbols and the system of geodesics of pseudo-Riemannian manifold (M, h = Hess g f ), establishes the relation between the components of the curvature tensors field of (M, h) and (M, g), and determines the PDEs representing the coincidence between the Christoffel symbols of (M, h) and the Christoffel symbols of (M, g). The last section presents the comparison of null-length curves trajectories obtained with the two metrics for a 5-dimensional RN black hole.
ABSTRACT The guest editors would like to express their gratitude to the reviewers for their serio... more ABSTRACT The guest editors would like to express their gratitude to the reviewers for their serious evaluation of the papers submitted in this special issue. Thanks also go to the authors for their interesting papers. Carlo Bianca acknowledges the support by the FIRB project RBID08PP3J (Metodi matematici e relativi strumenti per la modellizzazione e la simulazione della formazione di tumori, competizione con il sistema immunitario, e conseguenti suggerimenti terapeutici).
With the end positions of the stroke and the radius of the crank specified, the locus of the fixe... more With the end positions of the stroke and the radius of the crank specified, the locus of the fixed end of the crank in the plane of crank rotation can be graphically or analytically determined.
Section 1 recalls some basic notions of the geometry of jet bundles. Section 2 analyses the conne... more Section 1 recalls some basic notions of the geometry of jet bundles. Section 2 analyses the connection between least squares problem for ODEs and the geometric dynamics. Section 3 discusses the method of variational integrators for the Lagrangian dynamics and adopts a simplified ...
This paper investigates the way in which geometric dynamics on Riemannian manifolds can be applie... more This paper investigates the way in which geometric dynamics on Riemannian manifolds can be applied to fluid flow. The equations of motion from fluid mechanics (momentum equations), expressed on general curvilinear coordinates, are compared with the equations of motion in a gyroscopic field of forces. Attempts are made to seek conditions under which both describe the same motion, especially in the case when trajectories are not field lines. It is proved that all pathlines and streamlines are geodesics in the least square sense and, in some circumstances, other trajectories included in geometric dynamics have physical support.
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