nLab tangential notions of Frölicher spaces

Tangential Topics of Frölicher Spaces

This section is essentially lifted from Comparative Smootheology as I intend to remove it from that article and this seems a good place to put it.

Nearly every construction in differential topology starts with tangent or cotangent bundles. We shall look at how one could define these in the more general setting. An advantage of the dual nature of the definition of a Frölicher space is that just about every definition of a tangent or cotangent vector is possible. Essentially, as we have access to both curves and functionals we can consider push-forward?s and pull-back?s.

The ideas in this section can be traced at least as far back as Frölicher’s work in MR842916. That there are different orders of tangent and cotangent vectors appears in Kriegl and Michor’s book MR1471480, though it doubtless has antecedents. (However in MR1471480, only operational tangent vectors have higher orders. This is because the context is that of manifolds and so non-trivial higher order kinematic tangent vectors do not appear.)

Let us start with kinematic tangent vectors. We start with a notion of what it means for a curve to be flat.

Definition

Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space. A curve $c \in \mathcal{C}$ is said to be $k$-flat at $t \in \mathbb{R}$ if for each $f \in \mathcal{F}$, $(f c)^{(j)}(t) = 0$ for $1 \le j \le k$. It is flat at $t \in \mathbb{R}$ if it is $k$-flat for all $k \in \mathbb{N}$.

We write $\mathcal{C}_{x,k}$ for the subset of $\mathcal{C}$ consisting of those curves that map $0$ to $x$ and are $(k-1)$-flat at $0$.

For convenience, in the next definition we will say that any curve is $0$-flat.

Definition

Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space. Let $x \in X$ and let $k \in \mathbb{N}$. The $k$th kinematic tangent set at $x$, $T_{x,k} X$, is the quotient of $\mathcal{C}_{x,k-1}$ by the equivalence relation $c_1 \simeq c_2$ if $(f c_1)^{(k)}(t) = (f c_2)^{(k)}(t)$ for all $f \in \mathcal{F}$.

We write this as $T_{x,k} X$ rather than $T_x^k X$ to avoid conflict of notation with iterated tangent spaces. Since any curve can be composed with a suitable smooth map $\mathbb{R} \to \mathbb{R}$ to produce a curve of higher flatness, there are natural inclusions $T_{x,k} X \to T_{x,k+1} X$. For Euclidean spaces with their usual structure, all of these tangent sets coincide.

Definition

Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space. The full kinematic tangent set of $X$ at $x$ is $T_{x,\infty} X = \bigcup_k T_{x,k} X$.

It is obvious that each of these kinematic tangent sets is functorial in Frölicher spaces.

One would anticipate that in a given application the only two candidates for a kinematic tangent set will be either the full kinematic tangent set or the first kinematic tangent set. Note that for a manifold with boundary, the first kinematic tangent set at a point in the boundary is the tangent space of the boundary whereas the full kinematic tangent set is the tangent space of an ambient manifold (including the outward pointing normal).

We have been careful in writing “tangent set” rather than “tangent space” so as not to imply any particular structure. Reparametrisation of paths easily defines the structure of an $(\mathbb{R},+)$-set on each tangent set (scalar multiplication) but in general one will not be able to add tangent vectors. Nonetheless, some addition may be possible and addition has nice properties when it is defined.

Definition

Let $X$ be a Frölicher space, $x \in X$. For $u,v, w \in T_{x,k} X$ we say that $w$ is a sum of $u$ and $v$ if there are representatives $\alpha$, $\beta$, $\gamma$ such that $(f \alpha)^{(k)} = (f \beta)^{(k)} + (f \gamma)^{(k)}$ for all $f \in F_X$.

More generally, $u_1, \dots, u_k \in T_{x,k} X$ have a sum if there is some $w \in T_{x,k} X$ with $(f \beta)^{(k)} = \sum (f \alpha_i)^{(k)}$.

Lemma

Sums are unique if they exist.

Proof

If $\beta$ and $\gamma$ both represent sums of $u_1, \cdots, u_k$ then

(1)$$(f \beta)^{(k)} = \sum (f \alpha_i)^{(k)} = (f \gamma)^{(k)}$$

so $\beta$ and $\gamma$ represent the same vector in $T_{x,k} X$.

The construction of kinematic tangent vectors suggests a similar construction for cotangent vectors. As with tangent vectors, we need an auxiliary definition of flatness.

Definition

Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space, $x \in X$. A functional $f \in \mathcal{F}$ is said to be $k$-flat at $x \in X$ if for each $c \in \mathcal{C}$ with $c(0) = x$, $(f c)^{(j)}(0) = 0$ for $1 \le j \le k$. It is flat at $x \in X$ if it is $k$-flat for all $k \in \mathbb{N}$.

We write $\mathcal{F}_{x,k}$ for the subset of $\mathcal{F}$ of functionals that are $k$-flat at $x$.

Again, for convenience we will say that all functionals are $0$-flat.

Definition

Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space, $x \in X$. The $k$th kinematic cotangent space at $x$, $T^*_{x,k} X$, is the quotient of $\mathcal{F}_{x,k-1}$ by the relation $f_1 \simeq f_2$ if $(f_1 c)^{(k)}(0) = (f_2 c)^{(k)}(0)$ for all $c \in \mathcal{C}$ with $c(0) = x$.

The same discussion for kinematic tangent vectors applies to kinematic cotangent vectors except for the fact that cotangent vectors automatically form a vector space; sums are always defined.

Definition

The full kinematic cotangent space of $X$ at $x$ is $T^*_{x,\infty} X = \bigcup T^*_{x,k} X$.

There are obvious pairings between kinematic tangent and cotangent vectors based on evaluation. One defines a map $(f,c) \mapsto (f c)^{(k)}(0)$ and this descends to providing one knows that $(f c)^{(j)}(0)$ vanishes for $j \lt k$. The simplest case of this is where one of the tangent or cotangent vectors has order $k$. Thus the highest level pairing, $(f, c) \mapsto (f c)'(0)$, defines a pairing for all tangent and cotangent vectors but one which vanishes unless both are of first order.

We can also define operational tangent vectors. Recall that $\mathcal{F}$ is an algebra of functions.

Definition

Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space, $x \in X$. An operational tangent vector at $x$ is a derivation at $x$ of $\mathcal{F}$. That is, a linear map $\partial : \mathcal{F} \to \mathbb{R}$ such that $\partial(f g) = f(x) \partial g + g(x) \partial f$.

We write $D_x X$ for the vector space of operational tangent vectors at $x$ and $D_{x,k} X$ for the space of operational tangent vectors at $x$ that vanish on $\mathcal{F}_{x,k}$.

The order of an operational tangent vector is defined to be the least $k$ for which it lies in $D_{x,k} X$, otherwise it is said to have infinite order. Again, in a given application it may be preferable to restrict to operational tangent vectors of finite order, or of order $1$.

Lemma

There are natural maps $T_{x, k} X \to D_{x,k} X$ compatible with the connecting maps on each side.

Proof

For $c \in \mathcal{C}_{x,k-1}$ and $f \in \mathcal{F}$ we define an operational tangent vector by $f \mapsto (f c)^{(k-1)}(0)$. This has the required properties.

This map, however, need not be injective and neither need it map to a spanning set.

One can define two more versions of cotangent vectors by taking linear duals of the two versions of tangent vectors (this has to be done with care for the kinematic tangent vectors); however, these definitions may be thought of as one step removed from the smooth structure itself.