Jul 8, 2023: 50 ways to be a 2-sphere

It’s been a while (oops). I’ve went to a workshop on homological mirror symmetry, I moved, I’m now in France for a month, a lot has happened. But with a long train ride on front of me, I’m synthesizing what I’ve learned about geometric representation theory. It always comes back to examples, and the my favorite example is the two-sphere $S^2$. There’s many ways to think about this critter, and I want to list some. Even though they may seem horribly abstract, each of these is useful from some perspective. So I present to you, to the tune of a certian Paul Simon song:

50 ways to be a two-sphere

As a locus of points: $S^2$ is The set of points of unit Euclidean distance from the origin in $\mathbb{R}^3$
1. The classic. The most familiar notion, though it is very extrinsic.
Through real algebraic geometry: $S^2$ is the zero locus of the real quadradic equation, $x^2+y^2+z^2=1$.
As a topological manifold: $S^2$ is the genus 0 surface
1. Equivalently, we could say: $S^2$ is the only surface with no non-contractible loops, $S^2$ is the only surface with trivial first (co)homology
As a Riemannian manifold: $S^2$ is a surface with constant, positive gaussian curvature.
As a Symplectic manifold: $S^2$ is a genus zero surface with an area form with total area $4 \pi$
As a complex manifold: $S^2$ is the points with unit distance to the origin, together with the integrable almost complex structure $J:TS^2 \to TS^2$, defined by taking the cross product of a tangent vector with its base point (thinking of both as vectors in $\mathbb{R}^3$)
As a Kahler manifold: $S^2$ is a genus zero surface with complex structure as defined above, with Kahler form $\omega$ defined by $\omega(v,w) = \vert v \times w \vert$, thinking of $v,w$ as vectors in $\mathbb{R}^3$
As a space of directions: $S^2$ is the space of directions a vector in $\mathbb{R}^3$ can point
1. equivalently, it is the 2-fold (universal) covering space of $\mathbb{R}P^2$.
As complex projective space: $S^2 \cong \mathbb{C} P^1$, the space of one-dimensional complex subspaces of $\mathbb{C}^2$.
As a quotient space: $\mathbb{C} P^1$ is the quotient of $\mathbb{C}^2 - {0}$ by $\mathbb{C}^*$, the group of nonzero complex numbers which acts on $\mathbb{C}^2$ by multiplying each coeficent.
As a GIT quotient: $\mathbb{C} P^1$ is the projective GIT quotient of $\mathbb{C}^2$ by $\mathbb{C}^*$
As a symplectic quotient: $S^2$ is the symplectic quotient of $\mathbb{C}^2$ by $U(1)$, which acts by rotating each factor of $\mathbb{C}^2$.
1. This is equivalent to the the GIT quotient by the Kampf-Ness theorem
As the base of the hopf fibration: $S^2$ is the base of the hopf fibration, with fiber $S^1$ and total space $S^3$
1. this comes from the symplectic qoutient construction: The fiber $S^1$ is the group $U(1)$, and the total space is the preimage of the moment map.
As quaternions: $S^2$ is the space of unit quaternions, up to overall scaling by unit complex numbers
1. this is equivalent to the hopf fibration charecterization
In convex geometry: $S^2$ is the maximal volume constant-width surface in $\mathbb{R}^3$ with given diameter.
1. The inverse question (finding the minima;-volume constant width surface) is still open.
In Ricci flow: the ricci flow for any metric on $S^2$ converges to the round metric. That is, the round sphere is a Ricci soliton.
As a real symmetric space: $S^2$ is the coset space $SO(3)/U(1)$.
1. Equivently, the group of isomoetries of $S^2$ is $SO(3)$, which acts transitivly.
As a complex symmetric space: $\mathbb{C} P^1$ is the coset space $SL(2,\mathbb{C})/\mathbb{C}^*$.
as a flag manifold: $\mathbb{C} P^1$ is the space of full flags in $\mathbb{C}^2$.
As an abstract flag manifold: $\mathbb{C}P^1$ is the space of borel subgroups of $SL(2,\mathbb{C})$
As a coadjoint orbit: $S^2$ is a coadjoint orbit of $SO(3)$ on its dual lie algebra, $\mathbb{R}^3$.
As a scheme: $\mathbb{C} P^1$ is the spectrum of the ring of homogenous functions of degree 1 in 2 variables.
As a tautology: $S^2$ is the moduli space of points on $S^2$
As a functor of points: $\mathbb{C} P^1$ is a functor which sends a scheme $Y$ to the space of meromorphic functions on $Y$ (thought of as algebraic maps $Y \to \mathbb{C} P^1$ )
As a real moduli space: $S^2$ is the moduli space of closed quadralaterals in $\mathbb{R}^3$, modulo rigid rotations
As a complex analytic moduli space: $\mathbb{C} P^1$ is the moduli space of 4 points on $\mathbb{C} P^1$, modulo diagonal action by $SL(2,\mathbb{C})$.
In PDEs: $S^2$ is the dispersion relation of the 3-dimensional helmholtz equation
In gravity: $S^2$ is the isopotentizl surface of the laplace equation in 3 dimenions
1. in other words, spheres are the shapes of planets, the shape of black holes, etc.
As a toric manifold: $S^2$ is a cylinder with the top and bottom collapsed to a point

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