Geometric Dynamics on Riemannian Manifolds

Abstract

The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows.

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

“…is an analog of (2), where S is replaced by the mixed scalar curvature S mix , see (9), for the affine connection∇ = ∇ + T. The physical meaning of ( 4) is discussed in [2] for the case of T = 0. Our action (4) can be useful for the multitime Geometric Dynamics, e.g., [17] and survey [31]. This was introduced like Multi-time World Force Law involving field potentials, gravitational potentials (components of the two Riemannian metrics), and the Yang-Mills potentials (components of the Riemannian connections and the nonlinear connection).…”

Flow, Wind, and Stochastic Connectivity Modeling Infectious Diseases

“…e geometric data of the world [9,10] change the epidemic flow into an epidemic wind. is is a new idea that we are adding to the spread of infections.…”

“…ese are fundamental to understand the course of the epidemics and to plan effective control strategies for answering the question: how can we explain an exponentially growing number of patients all over the world who were diagnosed with COVID-19? e time has come for us to treat the epidemics like winds (geometric dynamics and geodesic motion in a gyroscopic field of forces) [9,10], producing chaotic dynamics. e geometric dynamics is generated by primordial data: flow and geometry of the space.…”

“…e mathematical literature that helped us to do this study is classified as follows: stochastic modeling of geometric structures [1][2][3][4], infectious disease flow [5][6][7], differential inclusions [8], geometric dynamics on Riemannian manifolds [9,10], nonholonomic optimization [11,12], and nonholonomic spaces [13].…”

Riccati PDEs That Imply Curvature-Flatness

“…Differential geometry is often considered "the art of manipulating partial differential equations (PDEs)". This point of view can be found in the papers [1][2][3][4][5][6][7][8][9][10][11][12], which develop the following topics: general theory of PDEs, symmetries and overdetermined systems of PDEs, fully nonlinear equations on Riemannian manifolds with negative curvature, basic evolution PDEs in Riemannian geometry, foundations of differential geometry, affine differential geometry, overdetermined systems of linear PDEs, geometric dynamics on Riemannian manifolds, the role of PDEs in differential geometry, the Dirichlet problem for first-order PDEs, and differential inclusions.…”