Differential Geometry as a Stannic Phenomenon

This post bears no mathematical fruit whatsoever, and is a somewhat random crazy thought. But I think blogs are the perfect setting for crazy ideas, so here we go.

The hope is to present a general framework for every type of differential geometry that has been studied. As presented here, we are definitely missing examples (for example geometries which involve certain multi-valued sections), but anything I could think of can be handled by minor modifications. We will make a minor remark about the example of locally conformal symplectic geometry towards the end of the post. On the other hand, I would be interested if there is any example of a geometry which lies wildly outside of what I present here. These are really naïve thoughts, so I'm sure any mathematician who stumbles across this post will have either something interesting to say or a philosophical description of why what I write is nonsense. Either is welcome in the comments!

In order to get this framework, we will be motivated by the fact that we should have a coordinate invariant description for the geometry involved. There will be four choices in such a description:

1. $n$, a choice of dimension for our underlying manifold
2. $\mathfrak{T}$, which tells us something about the underlying algebro-topological data (the letter T is for topological)
3. $\mathfrak{I}$, which is some choice of differential relations (the letter I is conditions of integrability)
4. $\mathfrak{N}$, yielding some sort of equivalence relation between the various choices of algebro-topological data satisfying the differential relations (the letter N is chosen because for many (but not all) examples, it comes from a normal subgroup of $\mathrm{GL}(n)$)

Correspondingly, a pair $(M,[D])$ consisting of a smooth manifold $M^n$ together with algebro-topological data $D$ (for data) living in $\mathfrak{T}$, solving $\mathfrak{I}$, and considered up to the equivalence defined by $\mathfrak{N}$, is to be called a $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$-manifold.

As the letters $\mathfrak{T}$, $\mathfrak{I}$, and $\mathfrak{N}$ spell out the chemical element TIN, and as there are four pieces of data specified when we also include the dimension, we shall refer to the study of $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$-manifolds as stannic geometry.

Let us make the data precise. We work over the ground field $\mathbb{R}$. For any fixed finite-dimensional vector space $V$ of dimension $n$, consider the massive vector space $$\mathcal{A}(V) := \left(\mathcal{T}(V) \otimes \mathcal{T}(V^*)\right),$$ where $\mathcal{T}(V) = \bigoplus_{j=0}^{\infty}V^{\otimes j}$ is the tensor algebra. We think of an element of $\mathcal{A}(V)$ as a single piece of data, and in order to allow ourselves arbitrary (but finite) amounts of data, we consider $$\mathcal{A}^{\infty}(V) := \bigoplus_{j=1}^{\infty} \mathcal{A}(V).$$

There is naturally a $\mathrm{GL}(V)$ action inherited on $\mathcal{A}(V)$, and hence on $\mathcal{A}^{\infty}(V)$ by acting on each factor. The underlying algebro-topological data for a stannic description of a manifold is a $\mathrm{GL}(n)$ invariant subset $\mathfrak{T} \subset \mathcal{A}^{\infty}(\mathbb{R}^n)$. (It is typical for $\mathfrak{T}$ to be an orbit set.) Given any isomorphism $V \cong \mathbb{R}^n$, we have an association $\mathcal{A}^{\infty}(\mathbb{R}^n) \cong \mathcal{A}^{\infty}(V)$, and the image of the subset $\mathfrak{T}$ is invariant under the choice of isomorphism. Hence, we may write simply $\mathfrak{T}(V) \subset \mathcal{A}^{\infty}(V)$.

Suppose $M$ is any manifold. Then we may consider the infinite-dimensional bundle $\mathcal{A}^{\infty}(TM)$ over $M$, where the fiber over $p \in M$ is just $\mathcal{A}^{\infty}(T_pM)$. We denote by $\Gamma(\mathcal{A}^{\infty}(TM))$ smooth sections of this bundle. (Note that even though the bundle is horribly infinite-dimensional, we take only smooth sections living in a finite-dimensional sub-bundle since we use direct sums but not direct products, and so the notion of smoothness is legitimate.) We represent by $\Gamma(\mathfrak{T}(TM))$ those sections $\omega$ such that $\omega_p \in \mathfrak{T}(T_pM)$ for each $p \in M$. An element of $\Gamma(\mathfrak{T}(TM))$ will be called an $\mathfrak{T}$-section on $M$. The data $D$ specified in a $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$ manifold is taken to be a $\mathfrak{T}$-section.

Remark: The fact that $\mathfrak{T}$ is $\mathrm{GL}(n)$-invariant is important, because $T_pM$ does not have a canonical trivialization at any point $p$.

Examples: A number of examples are certainly warranted. In all but the last of these examples, we take $$\mathfrak{T} \subset \mathcal{A}(\mathbb{R}^n) \cong \mathcal{A}(\mathbb{R}^n) \oplus \bigoplus_{j=2}^{\infty} 0 \subset \mathcal{A}^{\infty}(\mathbb{R}^n);$$ in other words, they are specified by one piece of data, which we simply take to lie in the first factor. The last example has $n$ pieces of data, and we may similarly fill up the first $n$ slots of $\mathcal{A}^{\infty}(\mathbb{R}^n)$ with these pieces of data.

1. If we take $\mathfrak{T}(V) = S^2_+(V)^*$ to consist of all positive-definite symmetric $2$-tensors, then a $\mathfrak{T}$-section is just a choice of Riemannian metric. Indeed, if $\mathfrak{I}$ and $\mathfrak{N}$ are trivial, then the corresponding stannic geometry is just Riemannian geometry.
2. If we have $n$ is even and take $\mathfrak{T}(V) = \Lambda^2_{\mathrm{non-deg}}(V)^*$ to consist of all non-degerate antisymmetric $2$-tensors, then a $\mathfrak{T}$-section is called an almost symplectic structure, and the corresponding stannic geometry is called almost symplectic geometry.
3. If again $n$ is even and we take $\mathfrak{T}(V) = \mathcal{J}(V) \subset V \otimes V^*$ to consist of consist of those elements which square to the $-I$ when naturally identified with elements of $\mathrm{End}(V)$, then we obtain almost complex geometry.
4. If we take $\mathfrak{T}(V) = V \setminus \{0\}$, then a $\mathfrak{T}$-section is called a non-zero vector field.
5. If we take $\mathfrak{T}(V) = \mathrm{Fr}_n(V) \subset \bigoplus_{j=1}^{n} V$, consisting of the set of $n$-frames, then a $\mathfrak{T}$-section is a parallelization of the tangent bundle.

Remark: If we stopped here, we would be stopping at a level of pure algebraic topology, which is interesting in its own right, but rather well understood. For example, we can perform a thorough obstruction-theoretic study for any of these examples to determine when a given manifold can be endowed with one of these structures.

1. Every manifold admits a Riemannian metric.
2. Any manifold admits an almost symplectic structure if and only if it admits an almost complex structure, in which case...
3. There are obstructions to admitting an almost complex structure which are well understood in low dimensions.
4. The existence of a non-zero vector field is equivalent to having Euler characteristic zero.
5. The Stiefel-Whitney classes are the primary obstructions to being parallelizable, although the question is surprisingly subtle. Every oriented $3$-manifold is parallelizable. Meanwhile, the only spheres which are parallelizable are $S^1$, $S^3$, and $S^7$, despite the fact that all spheres have trivial Stiefel-Whitney classes.

Our next step is to introduce the algebra of natural operations, which is $$\mathcal{B} = \mathrm{Hom}_{\mathrm{Diff}(\mathbb{R}^n)}(\Gamma(\mathcal{A}^{\infty}(T\mathbb{R}^n)),\Gamma(\mathcal{A}^{\infty}(T\mathbb{R}^n))),$$ where the diffeomorphism group acts naturally. Relatedly, we may consider the left $\mathcal{B}$-module of natural operations on $\mathfrak{T}$-sections, given by $$\mathcal{B}_{\mathfrak{T}} := \mathrm{Hom}_{\mathrm{Diff}(\mathbb{R}^n)}(\Gamma(\mathfrak{T}(T\mathbb{R}^n)),\Gamma(\mathcal{A}^{\infty}(T\mathbb{R}^n))).$$ The module structure is given in the obvious way. We require the elements of these structures to be smooth but not necessarily linear. A few examples are warranted to indicate that this is a reasonable notion of a natural operation (the last being a good example of non-linearity).

Examples:

• The exterior differential $d_k \colon \Gamma(\Lambda^k(T^*\mathbb{R}^n)) \rightarrow \Gamma(\Lambda^{k+1}(T^*\mathbb{R}^n))$ is a natural operation.
• The Lie bracket $[\cdot, \cdot] \colon \Gamma(T\mathbb{R}^n \oplus T\mathbb{R}^n) \rightarrow \Gamma(T\mathbb{R}^n)$ is a natural operation.
• Remark: Notice that it the Lie bracket does not define a natural operation $\Gamma(T\mathbb{R}^n \otimes T\mathbb{R}^n) \rightarrow \Gamma(T\mathbb{R}^n)$. There is a Schouten-Nijenhuis bracket on alternating multivector fields, which is a natural operation, but I'm less familiar with this.
• For any Riemannian metric $g$, there is a coordinate-invariant way of associating a Hodge star $\star_g$, yielding a natural operation on $S^2_+$-sections $\star_{\bullet} \colon \Gamma(S^2_+(T^*\mathbb{R}^n)) \rightarrow \Gamma(\mathcal{A}(T\mathbb{R}^n))$. Where exactly it lands in this image is a little complicated. We may decompose it into elements of $\mathrm{Hom}(\Gamma(\Lambda^k\mathbb{R}^n), \Gamma(\Lambda^{n-k}\mathbb{R}^n)) \cong \Gamma(\Lambda^k\mathbb{R}^n \otimes \Lambda^{n-k}(\mathbb{R}^n)^*)$, and then we may extend each of these to an element of $\Gamma(\mathcal{A}(T\mathbb{R}^n))$. Finally, we take the direct sum over $k$ to end at an element of $\mathcal{A}^{\infty}(T\mathbb{R}^n)$. Notice that $\star_{\bullet}$ is very highly nonlinear, and cannot even be continuously extended to $0 \in S^2(T^*\mathbb{R}^n)$. In particular, it is not simply the restriction of a natural operation in $\mathcal{B}$. It is also not a natural automorphism, since its image doesn't land in $S^2_+$-sections.

Remark: We may think of the algebras $\mathcal{B}_{\mathfrak{T}}$ as being a sheaf on $\mathcal{A}^{\infty}(\mathbb{R}^n)$ with topology defined by letting the open sets be $\mathrm{GL}(n)$-invariant subsets. The last example shows that this sheaf is not flasque.

It turns out that $\mathcal{B}$ is actually a little bit too big, in that it may act on a zero element in a nontrivial way. We instead will restrict our attention to those elements $\mathcal{B}_0 \subset \mathcal{B}$ which send the zero section to itself, and think of $\mathcal{B}_{\mathfrak{T}}$ therefore as a left $\mathcal{B}_0$-module.

Exercise: If $\phi \in \mathcal{B}$, then $\phi(0) = c = (c_1,\ldots,c_k,0,0,\cdots)$ where each $c_i$ is just a constant section (landing fiberwise in $\mathbb{R} \subset \mathcal{A}(T_p\mathbb{R}^n)$).

From the built-in naturality, we may now extend the operations of the various $\mathcal{B}_{\mathfrak{T}}$ to act on $\mathfrak{T}$-sections on an arbitrary manifold $M$. That is, we obtain a natural action $$\mathcal{B}_{\mathfrak{T}} \times \Gamma(\mathfrak{T}(TM)) \rightarrow \Gamma(\mathcal{A}^{\infty}(TM))$$ given by acting on local coordinate patches and using the built-in diffeomorphism invariance to glue the results. We may now fix a left $\mathcal{B}_0$-submodule $\mathcal{I} \subset \mathcal{B}_{\mathfrak{T}}$. Given an element $D \in \Gamma(\mathfrak{T}(TM))$, we may now ask whether $\mathcal{I}$ annihilates $D$. This condition is what one may refer to as an integrability condition.

Definition: Given a $\mathfrak{T}$-section $D$ is said to be $\mathfrak{I}$-integrable if $\mathfrak{I} \cdot D = 0$.

Remark: Notcie that a $\mathcal{B}$-submodule cannot annihilate any section, by the exercise above. It is for this reason that we restrict to $\mathcal{B}_0$-submodules. Typically $\mathcal{I}$ will be finitely generated, and this is all just a fancy way to say that each generator annihilates $D$.

Examples:

1. Suppose $\mathfrak{T} = S^2_+V^*$ so that a $\mathfrak{T}$-section is just a Riemannian metric. Let $\mathfrak{I}$ be the left sub-module generated by the natural operation on Riemannian metrics given by forming the Ricci tensor. Then if $\mathfrak{N}$ is trivial, we are simply studying Ricci-flat geometry.
2. If $n$ is even and we take $\mathfrak{T}(V) = \Lambda^2_{\mathrm{non-deg}}V^*$ as in almost symplectic geometry, then if we let $\mathfrak{I}$ be generated by the exterior differential $d$, then we are studying symplectic geometry.
3. For an almost complex structure, we may take $\mathfrak{I}$ to be generated by the object which forms the Nijenhuis tensor, and then we are studying complex geometry.
4. I don't believe there are any interesting natural operations on a single vector field, and so there's not much to do for the example when $\mathfrak{T}(V) = V \setminus \{0\}$.
5. One interesting thing to do in the parallelizable example is to ask that all of the frame given by the vector fields $(X_1,\ldots,X_n)$ satisfies that each Lie bracket $[X_i,X_j] = 0$. Then the underlying manifold comes with an $\mathbb{R}^n$-action given by translation along the given vector fields, and so if the manifold is connected, it is a quotient of $\mathbb{R}^n$ by a discrete subgroup, and hence of the form $(S^1)^k \times \mathbb{R}^{n-k}$.

Remark: By construction, and as these examples demonstrate, integrability conditions are local conditions.

When we defined $\mathcal{B}_{\mathfrak{T}}$, we simply replaced the domain of the relevant hom-space with $\mathfrak{T}$-sections. We may also replace the codomain. In particular, we may consider the natural automorphisms of $\mathfrak{T}$-sections, given by the group $$\mathcal{B}_{\mathfrak{T}} := \mathrm{Hom}_{\mathrm{Diff}(\mathbb{R}^n)}(\Gamma(\mathfrak{T}(T\mathbb{R}^n)),\Gamma(\mathfrak{T}(T\mathbb{R}^n))).$$ Our final piece of data is simply a subgroup $\mathfrak{N} \leq \mathcal{B}_{\mathfrak{T}}^{\mathfrak{T}}$. We define two $\mathfrak{T}$-sections $D$ and $D'$ to be $\mathfrak{N}$-equivalent if there exists some $n \in \mathfrak{N}$ such that $n \cdot D = D'.$ This completes our definition of $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$-manifolds, which we reminder the reader consists of data $(M^n,[D])$, where $D$ is a $\mathfrak{T}$-section satisfying the integrability conditions $\mathfrak{I}$ and taken up to $\mathfrak{N}$-equivalence.

Remark: In this definition, if a $\mathfrak{T}$-section $D$ is $\mathfrak{I}$-integrable, it is not necessarily the case that $n \cdot D$ is $\mathfrak{I}$-integrable. We will see this in the following example.

Example: There is one typical class of examples, if $N \unlhd \mathrm{GL}(n)$ is a normal subgroup, then we may declare two $\mathfrak{T}$-sections $D,D'$ to be $\mathfrak{N}_N$-equivalent if fiber by fiber they differ by the action of some element $n \in N$. (The normality of $N$ implies that this notion is well-defined regardless of a trivialization of each tangent space.) One typical example is $N = \mathbb{R}^*$.

• In the case of Riemannian geometry, we obtain conformal (Riemannian) geometry, in which two metrics $g$ and $g'$ are considered equivalent if they differ by multiplication by a nonzero function. In dimension $n=2$, this is just the setting of complex analysis.
• In the case of symplectic geometry, for example, we obtain conformal symplectic geometry, which is data of the form $(M,[\omega])$, where $\omega$ is a symplectic form, and two symplectic forms $\omega$ and $\omega'$ are said to be equivalent if there is some nonzero function $f$ such that $\omega = f\omega'$. Notice that the only such $f$ which preserve the symplectic condition are those which are constant, exemplifying the previous remark.
  

Remark: It is actually much more natural to talk about locally conformal symplectic geometry, in which $\omega$ is allowed to be a multi-valued symplectic form, such that the local sections are all equivalent up to conformal change. With minor modifications of the framework so presented, we can incorporate this example, but we shall not worry about it.

Example: Here is a more complicated example, realizing contact geometry as a stannic geometry. In it, we also get a flavor for what natural automorphisms of $\mathfrak{T}$-sections might look like.

• $\mathfrak{T} \subset \Lambda^0(\mathbb{R}^{2n+1})^* \oplus \Lambda^1(\mathbb{R}^{2n+1})^* \oplus \Lambda^2(\mathbb{R}^{2n+1})^*$ as those $(f,\alpha,\beta)$ with $f\alpha \wedge \beta^n \neq 0$. This is almost contact geometry.
• $\mathfrak{I}$ is generated by taking $d\alpha-\beta$ on the relevant components, which gives what one might call "strict" contact geometry, in which the data of the contact form is important
• $\mathfrak{N}$ is generated by the following automorphisms:
• For $g$ a non-zero function, $n_g$ is defined by $$n_g \cdot (f,\alpha,\beta) := (gf,\alpha,\beta)$$
• The operation $n_*$ defined by $$n_* \cdot (f,\alpha,\beta) = (f,f\alpha,df \wedge \alpha + d\beta).$$
  

Finally, we would like to understand how these stannic geometries interact. There is a lot to say, but here are a few basic observations about how to possibly modify $\mathfrak{T}$, $\mathfrak{I}$, and $\mathfrak{N}$.

• First, consider a (not necessarily linear) element $\phi \in \mathrm{End}_{\mathrm{GL}(n)}\mathcal{A}(\mathbb{R}^n)$. Then $\phi(\mathcal{T})$ is a $\mathrm{GL}(n)$-invariant set. We may consider $\phi_f$ as the fiberwise action of $\phi$, so that $\phi_f \in \mathrm{Hom}_{\mathrm{Diff}(\mathbb{R}^n)}(\Gamma(\mathfrak{T}(T\mathbb{R}^n)),\Gamma(\phi(\mathfrak{T})(T\mathbb{R}^n)))$. Hence, we have an induced submodule of $\mathcal{B}_{\phi(\mathfrak{T})}$ given as $$\phi(\mathfrak{I}) = \{y \in \mathcal{B}_{\phi(\mathfrak{T})} \mid y \circ \phi_f \in \mathcal{I}\}.$$ It follows that if $(M,[D])$ is an $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$-manifold, then $(M,[\phi_f \cdot D])$ is a $(\phi(\mathfrak{T}),\phi(\mathfrak{I}),\mathfrak{N})_n$-manifold. (Notice that this is well-defined, since if $D$ is $\mathfrak{N}$-equivalent to $D'$, then $\phi_f \cdot D$ is $\mathfrak{N}$-equivalent to $\phi_f \cdot D'$.)
• Second, if $\mathfrak{J} \subset \mathfrak{I}$ is a smaller submodule of $\mathcal{B}_{\mathfrak{T}}$, then any $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$-manifold $(M,[D])$ is automatically a $(\mathfrak{T},\mathfrak{J},\mathfrak{N})_n$-manifold.
• If $\mathfrak{N}' \geq \mathfrak{N}$, then if $(M,[D]_{\mathfrak{N}})$ is a $(\mathfrak{T},\mathfrak{I},\mathfrak{N})_n$-manifold, then $(M,[D]_{\mathfrak{N}'})$ is a $(\mathfrak{T},\mathfrak{I},\mathfrak{N}')_n$-manifold

There are further notions of functoriality between these which change dimension, and hence we may consider submanifolds and so on. However, as this post is already long, we shall end here, and possibly post more thoughts at a different time.