Hypersurfaces and Manifolds

In multivariable calculus, we study line integrals and surface integrals, as well as volume integrals and various theorems involving these concepts, including for example the gradient and divergence theorems. We then learn that manifold calculus provides vast generalizations of these concepts and results. We try to understand and tease this out further in this article.

To define line and surface integrals, we use the notions of curves for line integrals and surfaces for surface integrals, which are defined via continuous parameterizations:

Definition. A curve is the image of a continuous $c:I \rightarrow \mathbb{R}^{n}$ , where $I$ is an interval in $\mathbb{R}$ .

Equivalently:

Definition. A curve is the image of a continuous $c:\lbrack 0,1\rbrack \rightarrow \mathbb{R}^{n}$ .

We can see that these are equivalent by the following: a curve by the second definition is automatically a curve by the first, and a curve by the first definition is a curve by the second by defining $c_{2}(x) = c_{1}\left( \frac{x - a}{b - a} \right)$ , where $a,b$ are the endpoints of $I$ and $c_{1},c_{2}$ are the $c$ ‘s in the first and second definitions respectively.

(Actually, if $I$ is not closed at both endpoints then this isn’t strictly true, but let’s just assume closed intervals for the sake of simplicity. I don’t think anything we discuss here really changes whether we have this restriction or not.)

Definition. A surface is the image of a continuous $c:I \times J \rightarrow \mathbb{R}^{n}$ , where $I,J$ are intervals in $\mathbb{R}$ , such that this image is not also a curve. Equivalently, a surface is the image of a continuous $c:\lbrack 0,1\rbrack^{2} \rightarrow \mathbb{R}^{n}$ such that this image is not also a curve.

(These definitions agree with Wikipedia as well — look at the most general “topological” definitions, excluding differentiability/smoothness and other typically added conditions.)

A natural generalization appears upon looking at these definitions: the $n$ -dimensional analogue in $\mathbb{R}^{m}$ would be the image $I$ of a continuous $c:\lbrack 0,1\rbrack^{n} \rightarrow \mathbb{R}^{m}$ such that $I$ is not the image of some other continuous $c^{'}:\lbrack 0,1\rbrack^{k} \rightarrow \mathbb{R}^{m}$ for some $k < n$ . As before, this is equivalent if we replace $\lbrack 0,1\rbrack^{i}$ by a product of $i$ intervals.

(Earlier, when first writing this on April 22, 2021, I had written the definition such that the domains of $c$ and $c^{'}$ could be any open sets $U$ and $U^{'}$ . However, based on comparing to the Wikipedia definitions as well as my intuition, I don’t think now that that makes sense. Thus, we’ll stick with products of intervals for domains.)

So far, I have not learned a term that is standardly used to describe this concept. Initially, when I first learned about the word “hypersurface,” I thought that would be the perfect word, suggesting a beyond-two-dimensions generalization of a surface. (Hence, the title of this post.) However, it seems that the standard meaning of this term is different: Wikipedia defines a hypersurface quite generally using manifolds, but also requires a hypersurface to be embeddable in a space of one higher dimension, which doesn’t necessarily seem always true for our concept. Thus, despite the name, even allowing for extra generality via manifolds, the term hypersurface doesn’t seem to be applicable to our concept. I thought about the term “parametric hypersurface” as well, since our definitions are based on continuous parametrizations; however, this term still wouldn’t be the most applicable, again because of the dimension requirement on hypersurfaces. Furthermore, my friend Akshay found the link https://math.stackexchange.com/questions/4204088/parametric-versus-non-parametric-hypersurface, which suggests that the term “parametric hypersurface” already has a different standard definition which doesn’t seem obviously equivalent. I’ll pick the term “hypercurve” here, as based on a quick Google search it doesn’t seem to have a standard meaning in math yet.

Now, the Wikipedia article on hypersurfaces states that a hypersurface in two-dimensional space is a plane curve, and a hypersurface in three-dimensional space is a surface. A natural generalization of this, using our terminology, would be:

Conjecture. A hypersurface of dimension $n$ is a hypercurve with $n = m - 1$ .

(When first writing this, for some reason I had put down $n = m$ ; this however doesn’t make sense to me now, so I’m not sure whether that was a mistake or if there was a reason for that which I don’t remember now.)

Not sure how we would prove this. Also, is this equivalency, or just that a hypersurface is a hypercurve but not necessarily vice versa? Since manifolds would seem to be vastly more general, and since hypersurfaces are defined via manifolds, it seems that equivalency could be possible here. This would then establish a hierarchy of generalizations: hypersurfaces generalize hyperplanes, plane curves, and surfaces (as stated in the Wikipedia article), while hypercurves generalize hypersurfaces. (The terminology would be a bit wonky then since intuitively “surfaces come after curves,” but this doesn’t seem to be too much of an issue since “hypercurve” doesn’t seem to have a standard definition yet.)

Now, manifold calculus is typically described as a vast generalization of constructions like these. Thus, we would expect the following to be true:

Conjecture. A hypercurve is a manifold.

If this isn’t true, then we need to see whether there are merits to the concept of a hypercurve separate from manifold calculus. However, from the discussion in Wikipedia’s article on surfaces, it does seem that manifolds are a vast generalization of hypercurves.

The main immediate issue in trying to prove this conjecture is that the parametrization $c$ isn’t assumed to be invertible or for such an inverse to be continuous; we only assume that $c$ is continuous.

Attempts to prove these follow.

We start with the first conjecture. Let $S$ be an $(n - 1)$ -dimensional hypersurface embedded in $\mathbb{R}^{n}$ , so $S \subseteq \mathbb{R}^{n}$ . (This change from embedding to inclusion produces no loss of generality with the technical topological definition of “embedding.”) Wikipedia asserts that every hypersurface is defined implicitly locally by a single equation

$\displaystyle f\left( x_{1},\ldots,x_{n} \right) = 0.$

If we show that such an equation implies parameterization, then we show that hypersurface implies hypercurve.

However, right now I don’t know the rigorous definition of “locally” and I would need to look this up. This would be an essential step in the proof.

To try to show the other direction (hypercurve with $n = m - 1$ is a hypersurface), we recall the general topological definition of “ $(m - 1)$ -dimensional manifold”: a second-countable Hausdorff topological space $M$ such that, for each $x \in M$ , $x$ is contained in a neighborhood $N(x)$ that is homeomorphic to $\mathbb{R}^{m - 1}$ . Actually, different definitions by different authors seem to vary a bit in generality, so we exclude the second-countable condition.

Given a hypercurve with parametrization $c$ (let’s say now that the domain of $c$ excludes endpoints — I don’t think this change causes problems at least for a proof sketch), we may select $N(x) = (0,1)^{m - 1}$ as well as $M = (0,1)^{m - 1}$ . We have that $c$ is continuous, but that doesn’t guarantee that it is invertible or a homeomorphism (it is in fact possible for $c$ to be invertible and continuous but not a homeomorphism if the inverse isn’t also continuous.) Thus, we probably need a little more work here.

First written April 22, 2021, but modified November 27, 2022 to better reflect standard terminology.

—

I’ll discuss some updates. On November 7, 2022, I learned about the Whitney embedding theorem. Taking the strong version (see the Wikipedia article), we have that any $n$ -manifold (assuming common conditions like Hausdorff and second-countable) can be embedded into $\mathbb{R}^{2n}$ . Before learning about this, I had considered a “relaxed” definition of a hypersurface as an $n$ -manifold that can be embedded into $\mathbb{R}^{m}$ for some $m \geq n$ ; however, assuming Hausdorffness and second-countability, this is exactly equivalent to a manifold. It thus makes sense to have the term “hypersurface” mean something more specific.

Also, it seems like I had misread the definition of a hypersurface where I didn’t realize that it could also be defined to be an algebraic variety, and according to the Wikipedia article there are cases of such hypersurfaces that aren’t manifolds; however, I consider this to be a different definition that belongs more to the subject of algebraic geometry, and thus here I will continue defining a hypersurface to be a manifold.

—

EDIT (01/23/2023): I read The_Theorems_of_Vector_Calculus.pdf (ucla.edu), which clarifies things a lot for me.

Let’s trace through how the “Fundamental Theorems of Multivariable Calculus” (Kelvin-Stokes Theorem, Divergence Theorem, etc.) are developed. We define curves, surfaces, and volumes – these are all special cases of hypercurves. Then, we respectively define line integrals, surface integrals, and volume integrals. In a similar way, we can imagine that we can define an integral over any hypercurve – we can define any “hypercurve integral.” Then, for a certain kind of surface $S$ (which is orientable and has a “nice enough” boundary and such), we have a theorem (the Kelvin-Stokes Theorem) which says that: for any “nice enough” function, the integral of a suitable derivative of the function over $S$ is equal to the integral of the function over the boundary of $S$ . (This last integral is a line integral.) The Divergence Theorem, of course, has a similar form, relating a suitable volume integral over interior and surface integral over boundary.

For hypercurves, the formulation would then look like: for a certain kind of hypercurve $C$ , we have a theorem (presumably an application of the generalized Stokes’ Theorem) which says that: for any “nice enough” function, the integral of a suitable derivative of the function over $C$ is equal to the integral of the function over the boundary of $C$ . This last integral would actually be a hypercurve integral itself (for a one-dimension-lower hypercurve), so we’d need to have that the boundary of a hypercurve is itself a hypercurve. This formulation doesn’t actually encompass the Gradient Theorem since a two-point boundary wouldn’t be a hypercurve; instead, the (seemingly more general) maxim of “integral over boundary is equal to integral of derivative over interior” does. I would guess that the “certain kind” requirement for the hypercurve is what guarantees it’s a manifold, so that this all falls under manifold calculus as we’d expect.

Thus, to summarize:

Just as we defined line integrals, surface integrals, and volume integrals, we can define an integral over any hypercurve.
With suitably imposed conditions on a hypercurve, analogous to those imposed on surfaces and volumes in the Curl and Divergence Theorems, the hypercurve is guaranteed to be a manifold, and then the generalized Stokes’ Theorem provides exactly the generalization of the Curl and Divergence Theorems that we’d expect.

These are our (very reasonable) expectations now, and I’d probably proceed to learn manifold calculus to actually see this.

I have another separate but possibly related question here, regarding Fubini’s Theorem. Separately from the generalized Stokes’ Theorem and the “boundary-derivative” maxim, another way to generalize the single-variable Fundamental Theorem of Calculus is via Fubini’s Theorem, which allows evaluation of double, triple, and multiple integrals. How does this relate to that? Can we express Fubini’s Theorem as a special case of generalized Stokes’ Theorem as well? In that case, I don’t know how the iterated integral would be interpreted as the “integral over a boundary,” or what the associated derivative would be. So maybe these are just two separate ways to generalize the single-variable Fundamental Theorem of Calculus?

Rectangles in $n$ -space are actually hypercurves (well, I haven’t written a formal proof that rectangles are hypercurves, but I’m pretty sure), so Fubini’s Theorem is basically about an integral over a hypercurve too, just a (potentially) different kind of hypercurve. Based on a quick Google search, it seems like Fubini’s Theorem and the generalized Stokes’ Theorem don’t show up much together, so they seem to be essentially generalizations of the single-variable Fundamental Theorem of Calculus in separate directions.

Actually, in light of this StackExchange question, I’m not sure anymore. I need to learn more — basically, I need to learn manifold calculus to figure this out. Either way, though, the organization of subjects would result nontrivially from theorems, not just from purposely-constructed definitions. Conceptually, I would now define manifold calculus to be the generalization of the “boundary-derivative” maxim exhibited by certain results in real multivariable calculus, where this could happen to also generalize other results in real multivariable calculus like Fubini’s Theorem.

If it turns out that Fubini’s theorem isn’t a part of manifold calculus, then that would imply that manifold calculus is not just a strict generalization of real multivariable calculus. Then, multivariable calculus would include results for calculus over certain hypercurves that the application of manifold calculus to $\mathbb{R}^n$ wouldn’t encompass. In fact, by the Whitney embedding theorem, that would imply that manifold calculus is really a special case of real multivariable calculus! Actually, the Whitney embedding theorem assumes some common conditions on the manifold, so if we want to take the most general definitions and avoid making those assumptions, then it would turn out that multivariable calculus and manifold calculus are separate subjects, which admittedly have a rich intersection. Then, I’d push back against the idea that the Gradient Theorem, Curl Theorem, Divergence Theorem, and Green’s Theorem are the Fundamental Theorems of Multivariable Calculus, as I’ve typically seen stated throughout the Internet, since that would ignore Fubini’s Theorem as another “fundamental theorem” in a separate direction.

But from the StackExchange answer, it looks like this isn’t true, and that Fubini’s Theorem does have a straightforward generalization to manifold calculus. If this is the case, and if manifold calculus does happen to encompass other generalizations too, then it could potentially be approached from other points of view or motivations — indeed, many other math subjects can be approached from multiple points of view. In that case, I would expand beyond just the “boundary-derivative” approach.

Share and like (need WordPress account to like):

Leave a comment Cancel reply