What is a manifold? #

Intuitively it is best to think of a manifold as a topological space with certain properties. The most important of which is that given any section of the space, the section overall looks similar to euclidean space. Which is what we refer to as being “locally euclidean”. This is similar to the notion that “locally” the Earth looks flat, but once we back up we realize that it is actually round. In fact almost every geometrical constructions can be thought of as a manifold.

Formally the definition of locally euclidean is simply the following:

Definiton

A topological space is locally euclidean if for every point in the space their exists a neighborhood which homeomorphic to the n-dimensional space over the real numbers with the usual topology ($\mathbb{R}^n$).

This sounds reasonable and intuitive but we need one more condition to make this a manifold, we need the entire space to be Hausdorff. For every student of topology this is reasonable enough, as we want some notion of separation between points, and this is usually the weakest notion of separation between points that agrees with our intuitive notions of such. So finally we can define a manifold thusly:

Definiton

A topological manifold is a topological space that is locally euclidean and Hausdorff.

What does it mean to be smooth? #

Naturally, we view this notion in the underlying topology, and the entire notion of smoothness doesn’t make much sense otherwise. Mathematically we view a function as “smooth” if it is differentiable along the entire domain. Since functions that are continuous everywhere not necessarily differentiable, but differentiable functions are continuous, we view the continuous functions as zero times differentiable. Once (continuously) differentiable functions would be the next step logical step. This naturally creates a hierarchy of differentiable functions. If we only allow functions that have derivatives on our topological structure, then we would also have a smooth topological structure. In order for this to make sense, we need to define a differential structure on the topology that is global.

We can trivially assign a differential structure to every topological manifold locally by applying a homeomorphism from the topology to a linear space, the hard part is defining one that is global. A through discussion of different differential structures is beyond the scope of this introduction, and for our purposes just assume we are dealing with the standard notions of differentiability.

Note that smoothness (and differential structures) are not guaranteed to exist in every topology that we define, so we need to be careful. We also can define isomorphisms between topologies, but this will not necessarily be enough to maintain the differentiability of functions. If we can guarantee this, we call this special form of isomorphism a diffeomorphism. So if we restrict the set of all topologies, to the set of all topologies that allow smoothness, and restrict the isomorphisms to only diffeomorphisms, we get the theory of smooth manifolds.

Putting these all together #

Definiton

A (finite dimensional) smooth manifold is a topology that is Hausdorff and locally Euclidean, with functions on the topology that obey some differential structure.

This definition seems simple enough but as with many things the devil is in the details.

Note that we are brushing against the theory of Lie groups and Lie algebras, as a Lie group is a group consisting of smooth manifolds. All Lie groups give rise to a Lie algebra through a standard construction, and conversely any finite dimensional Lie algebra (over the real or complex numbers) gives rise to a Lie group. Note that this does not hold in infinite dimensions, mirroring similar things that we come across later.

What are infinite dimensional manifolds? #

Taking all the previous information together, one might be interested in generalizing these notions from a theory in finite dimensions to a theory of over infinite dimensions. However we must be very careful since when we work over finite dimensional manifolds, and vector spaces, most of our standard notions of what a manifold should be are fairly intuitive.

As we move to infinite dimensions certain notions fragment and we need more assumptions in order to preserve the structures we have defined. There is no one type of infinite dimensional manifold since we need to model these manifolds on topological vector spaces (discussed below). The standard definition of manifold instantly breaks once we introduce infinite dimensions. The very notion of being locally euclidean requires us to embed the manifold in some subset of $\mathbb{R}^n$.

As we will see in a bit, even our standard notion of differentiability breaks fairly easily in an infinite dimensional context.

Topological Vector Space #

First we must briefly talk about topological vector spaces.

Definiton

A topological vector space is a topology over a field that preserves vector addition and scalar multiplication.

Note that if our underlying field was either the real or complex numbers, we naturally already have a topological vector space over our manifolds. However to start discussing the idea of infinite dimensional differentiability we need much more structure.

Generalizing to Infinite Dimensions. #

A smooth manifold on the topological vector space $\mathbb{E}$ is a Hausdorff topological space $\textbf{M}$ with a family of charts, $(\phi_\alpha, U_\alpha)_{\alpha \in A}$ such that:

• $U_\alpha \subseteq \textbf{M}$ are open sets.

• $\bigcup_{\alpha \in A} = \textbf{M}$. In other words, you can take a collection of open sets, that unioned together give us $\textbf{M}$.

• $\phi_\alpha: U_\alpha \rightarrow \phi_\alpha(U_\alpha)\subseteq\mathbb{E}$ are homeomorphisms. Note that the mapping is closed (open sets map to open sets).

• $\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \rightarrow \phi_\beta(U_\alpha \cap U_\beta)$ are appropriately smooth.

Vectors Spaces with more structure #

If we equip a topological vector space with more structure we get the following types of spaces: Hilbert, Banach, Fréchet and Convenient. These are listed in order from strongest to weakest, or from the least general to the most general. So lets define each of these.

Definiton

A Hilbert space is a topological vector space with an inner product, such that the norm defined by the square root of the inner product, of any element with itself is the magnitude of that element, and that norm turns the space into a complete metric space. In other words it is a complete inner product space.

Definiton

A Banach space is complete vector space with a norm.

Q: What’s yellow, linear, normed, and complete? A: A Bananach space.

Definiton

A Fréchet space is complete space locally convex space with some metric defined on it. In other words it has some notion of distance.

Definiton

A Convenient space is a vector space that is locally convex, and every Mackey-Cauchy sequence converges in the space. This is also known as locally Cauchy.

As you notice, each of these gets substantially weaker with Convenient spaces not being that convenient to work in. Naturally we would love to work in a Hilbert or a Banach space whenever we can, but as we will see this sometimes is not enough, and we will need more general spaces in order to fulfill certain conditions. Also keep in mind that none of these definitions have had any mention of the number of dimensions, finite or otherwise. Given this it is perfectly reasonable to talk about a finite dimensional Hilbert space, or an infinite dimensional Fréchet space.

Notions of differentiability in infinite dimensional spaces #

Note that everything we have discussed so far can be generalized trivially to infinite dimensions, however the main problem is that our notion of the derivative might not work in each of these cases. For example, in a Fréchet space there is no notion of a norm, so our definition can’t have anything remotely as strong. In this case we will use various generalizations of the derivative in order to truly understand what is going on. There are multiple ways to define a derivative, and it mostly depends on how many properties we have on our underlying space. Under a Banach space we have the most freedom, and can choose how we wish to define differentiation depending on which form is the most convenient. Most generalized notions of differentiability, such as Hadamard differentiability, and Carathéodory differentiability work in Banach spaces, but since we are not restricted to Banach spaces we will only define the two most common forms of differentiation, the Fréchet derivative and the most general form of differentiability which is the Gâteaux derivative.

Definiton

Given two complete normed vector spaces $V$ and $W$, and $f: V \rightarrow W$, and given a continuous linear operator $A: V \rightarrow W$, i.e. the linearization of $V$ at $W$, we can define the Fréchet derivative of $f$ at $U$, where $U$ is an open subset of $V$, by the linear map

$$\lim_{h \to 0} \frac{ \| f(x + h) - f(x) - Ah \|_{W} }{ \|h\|_{V} } = 0.$$

Definiton

Given two convenient vector spaces, $U$ and $V$, let $f: U \rightarrow V$ be a function between them. $f$ is Gâteaux-differentiable if there exists an operator $T_X: X \rightarrow Y$ such that for all $v \in X$:

$$\lim_{t \to 0} \frac{f(x+tv) - f(x)}{t} = T_Xv.$$ The operator $T_X$ is referred to as the Gâteaux derivative of $f$ at $x$.

Keep in mind that the Gâteaux derivative is strictly weaker than the Fréchet derivative as Gâteaux differentiability implies Fréchet differentiability, but the converse does not hold. Also if we take a Fréchet space, then the Gâteaux differential operator exists, and is bounded. However in general this does not hold, and the Gâteaux derivative doesn’t even have to be linear.

Breaking Intuition #

At the level of infinite dimensional manifolds we can easily break our intuitive notion of what should be true.

For example given a Fréchet space, let $C(\mathbb{R})$ be the space of continuous functions with the compact open topology. Define a mapping of functions

$$f: C(\mathbb{R}) \rightarrow C(\mathbb{R})$$ $$f(x)= e^x.$$

This function has a derivative, and its derivative is invertible everywhere, but the image does not have an inverse, because the set that consists of the inverse is not open in the topology. Therefore the inverse function theorem does not hold in general for Fréchet spaces. Therefore the inverse function theorem does not hold in Fréchet spaces.

Closing Remarks #

This is a light introduction, but this will help get you started on in the field of infinite dimensional manifold theory. This isn’t even scratching the surface and you can spend years studying this field before getting to the edge of knowledge. Keep in mind that most of this was written from the perspective of a mathematician, so significant portions of it will be distinctly different that vast swaths of the literature. This field is at an intersection of physics and mathematics, and frequently has contributions from both of them. From the mathematicians standpoint I have found that the vast amount of useful resources comes from Andreas Kriegl, Peter W. Michor, and their respective graduate students.

Be careful when using any resource (including this page) for too much as this is still an active area of research and things change regularly, definitions change from author to author, and I am not an expert in the field (though I quite enjoyed learning about it).

References #

Fréchet space. from Wolfram MathWorld. (n.d.). http://mathworld.wolfram.com/FrechetSpace.html

Locally Convex. from Wolfram MathWorld. (n.d.). http://mathworld.wolfram.com/LocallyConvex.html

Gâteaux Derivative. from Wolfram MathWorld. (n.d.). http://mathworld.wolfram.com/GateauxDerivative.html

Fréchet Derivative. from Wolfram MathWorld. (n.d.). http://mathworld.wolfram.com/FrechetDerivative.html

Carathéodory Derivative. from Wolfram MathWorld. (n.d.). http://mathworld.wolfram.com/CaratheodoryDerivative.html

Kriegl, A., & Michor, P. W. (1991). Aspects of the theory of infinite dimensional manifolds. Differential Geometry and Its Applications, 1(2), 159-176. doi:https://doi.org/10.1016/0926-2245(91)90029-9

Kriegl, A., & Michor, P. W. (1997). The convenient setting of global analysis. Providence, RI: American Math. Soc. Michor, P. W. (2017, May 28). Infinite dimensional manifolds. Retrieved December 9, 2018, from https://www.mat.univie.ac.at/~michor/Madrid-2016.pdf

Sakai, K. (2020). Fundamental results on infinite-dimensional manifolds. Springer Monographs in Mathematics, 81–201. https://doi.org/10.1007/978-981-15-7575-4_2