This question is motivated by the Lorenz curve used in economic analysis as well as by the Penrose diagram used in general relativity, used by physicists to visualize cause-and-effect relationships in Minkowski compacted space-time models. .

He is also deeply motivated by Hamiltonian mechanics, symplectic geometry and contact geometry (which can be considered the odd dimensional counterpart of symplectic geometry). Hamiltonian mechanics was a reformulation of Newtonian mechanics. In Hamiltonian mechanics, we study phase spaces of physical systems, symplectic flows, and many other subjects.

From a mathematical point of view, this question is motivated by modern trends in differential geometry, topology, abstract algebra and mathematical physics.

Consider families of flat curves for $ Re (s) ge1: $

$$ zeta: = {(x, y) in Bbb R ^ 2 | x ^ s + y ^ s = 1 } $$

$$ tau: = {(x, y) in Bbb R ^ 2 | (1-x) ^ s + y ^ s = 1 } $$

$$ psi: = {(x, y) in Bbb R ^ 2 | x ^ s + (1-y) ^ s = 1 } $$

$$ phi: = {(x, y) in Bbb R ^ 2 | (1-x) ^ s + (1-y) ^ s = 1 }. $$

Interpretation $ zeta, tau, psi, $ and $ phi $ because the phase spaces allow their interpretation as infinite dimensional varieties, in particular of infinite dimensional symplectic manifolds:$ ( zeta, omega), ( tau, omega), ( psi, omega), ( phi, omega).

The following is a natural Hamiltonian vector field, which defines a Hamiltonian flow on each of the symplectic manifolds.

The process of lifting these varieties into $ Bbb R ^ 3 $ can be realized with homotopic maps. See Paul Frost's answer here: https://math.stackexchange.com/questions/2895816/existence-of-homotopic-map. The essential point to understand is that each of the four symplectic manifolds is a unique projection of the curves recorded by homotopy. In other words, one can project the relief curves on the plane curves in a bijective way. The image provides the intuition of what a single elevator looks like. But in reality, there is an infinity of elevators at different heights above planar curves.

Questions $ ($choose one$) $:

$ 1) $ Can we stick together $ zeta, tau, psi, phi $ including all the elevators and treat it all as a multiple? What would be the complete classification of this variety?

$ 2) $ Can a $ 3 $– to be built, whose geodesics are projected on these planar algebraic curves?

I think so, because we can project all the phase-elevated spaces to the corresponding planar phase spaces. I'm pretty sure that the $ 3 $-manifold is a submultiple of the infinite dimensional variety.

$ 3) Define a surface that is an immersion of $ Bbb RP ^ 2 $ in $ Bbb R ^ 3, $ which looks like this, and is a three-dimensional analog of the two-dimensional equations above:

This is a form I have constructed to try to get an intuitive understanding of the geometry of variety and phase space in three dimensions:

The Boy's Surface is an example of immersion from $ Bbb RP ^ 2 $ in $ Bbb R ^ 3. $

The picture concerns the equations because each white strand of the image corresponds to a geodesic strand on a $ 2 $– various, namely $ S ^ 2. $ Each white geodesic also corresponds to a specific plane curve indicated above for a specific value of $ s $. In the photo $ s = 2. $ Singularities can be considered as invariants or fixed points because they do not change geographic location for the purposes of this question. variant $ s $ gauge the appearance of the form. As $ s $ approach to the infinite form will look like a cube. As $ s $ tends to $ 1, $ the shape will look like $ 2 $ pairs of perpendicular lines located in $ 3 $-space.

It may help to say that the image is the shape of the intersections of $ 8 $ identical copies of $ S ^ 2 $ with finely many geodesics shown. It's like a venn diagram of higher dimensions if you want.