This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in compactified Minkowski space time models.

It is also motivated deeply by Hamiltonian mechanics, symplectic geometry, and contact geometry (which can be viewed as the odd dimensional counterpart of symplectic geometry). Hamiltonian mechanics was a reformulation of Newtonian mechanics. In Hamiltonian mechanics one studies phase spaces of physical systems, symplectic flows and many other topics.

From a mathematical standpoint, this question is motivated from modern trends in differential geometry, topology, abstract algebra and mathematical physics.

Consider the families of planar 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 \}. $ $

Interpreting $ \zeta,\tau,\psi,$ and $ \phi$ as phase spaces allows the interpretation of them as infinite dimensional manifolds, specifically infinite dimensional symplectic manifolds:$ (\zeta,\omega),(\tau,\omega),(\psi,\omega),(\phi,\omega).$

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

The process of lifting these manifolds into $ \Bbb R^3$ can be achieved with homotopic maps. See the answer by Paul Frost here: https://math.stackexchange.com/questions/2895816/existence-of-homotopic-map. The key thing to understand is that each of the four symplectic manifolds are unique projections of the curves lifted via homotopy. In other words, one can project the lifted curves onto the planar curves bijectively. The image provides the intuition for what a single lift looks like. But in reality there are infinitely many of these lifts each at different heights above the planar curves.

Questions $ ($ choose one$ )$ :

$ 1)$ Can one glue together $ \zeta,\tau,\psi,\phi$ including all lifts and treat the whole thing as a manifold? What would be the full classification of this manifold?

$ 2)$ Can a $ 3$ -manifold be constructed, whose geodesics project to these algebraic planar curves?

I think yes, because one can project all lifted phase spaces down to the corresponding planar phase spaces. I’m pretty sure the $ 3$ -manifold is a submanifold of the infinite dimensional manifold.

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

This is a shape I built to try to get an intuitive understanding of the geometry of the manifold and phase space in three dimensions:

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

The image relates to the equations because each white strand in the image corresponds to a strand of a geodesic on a $ 2$ -manifold, namely $ S^2.$ Each white geodesic also corresponds to a specific planar curve listed above for a specific value of $ s$ . In the picture $ s=2.$ The singularities can be viewed as invariants or fixed points because they don’t change spatial location for the purpose of this question. Varying $ s$ gauges the way the shape looks. As $ s$ approaches infinity the shape will look like a cube. As $ s$ tends to $ 1,$ the shape will look like $ 2$ pairs of perpendicular lines situated in $ 3$ -space.

Maybe it helps to say that the image is the shape of the intersections of $ 8$ identical copies of $ S^2$ with finitely many geodesics shown. It is like a higher dimensional venn diagram if you will.