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Translation preserves an embedded disc and its tangent space

Statement

Let $0 \leq d \leq m$ be integers and let $r \in {1, 2, \dots} \cup {\infty}$ . For maps between positive-dimensional Euclidean open sets, $C^{r}$ has its componentwise meaning: every iterated coordinate partial derivative of order at most $r$ exists and is continuous, with every finite order required when $r = \infty$ . Maps with zero-dimensional source or target are declared $C^{r}$ ; their total derivatives are the unique linear maps in the corresponding zero-dimensional case. A $C^{r}$ embedded $d$ -disc here is the image $W = j (D)$ of a $C^{r}$ map $j : D \to R^{m}$ from a nonempty open ball $D \subseteq R^{d}$ that is injective, is a homeomorphism onto its image with the subspace topology, and has injective total derivative everywhere.

At $x = j (a)$ , identify tangent vectors with vectors of $R^{m}$ and set $T_{x} W = im D j (a)$ . This tangent subspace is independent of the admissible parametrization of the image disc.

Assume explicitly that $0 \in D$ and $j (0) = 0$ . For any $p \in R^{m}$ , the translated set $W_{p} = p + W$ is a $C^{r}$ embedded $d$ -disc through $p$ , parametrized by $j_{p} (a) = p + j (a)$ , and $T_{p} W_{p} = T_{0} W$ under these Euclidean identifications. The zero-dimensional disc is allowed and is a singleton with zero tangent subspace.

Statement references

The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder

Cᵏ Euclidean maps and diffeomorphisms

Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological

Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

Facts & Assumptions

[F1]

Total derivatives use the linear approximation with normwise remainder in The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder, including zero dimensions. The positive-dimensional $C^{r}$ convention is Cᵏ Euclidean maps and diffeomorphisms; for $r \geq 1$ it supplies $C^{1}$ regularity and continuous total derivative. The zero-dimensional convention is stated above.

[F2]

A homeomorphism is a continuous bijection with continuous inverse Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological. The image carries the subspace topology, whose open sets are traces of ambient open sets, and restrictions and corestrictions of continuous maps obey its characteristic property Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace.

[F3]

At a positive-dimensional immersion point, A Euclidean immersion is locally the canonical inclusion and is locally an embedding with $k = 1$ supplies local $C^{1}$ source and ambient coordinates in which the map is the inclusion $u \mapsto (u, 0)$ . Derivatives of composites satisfy The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a).

Proof

Given: The dimensions, differentiability order, embedding, explicit basepoint hypotheses and translation in the Statement.

Goal: Preserve embeddedness, dimension and the intrinsic tangent subspace.

1.1givenF1algebra

Define $j_{p} (a) = p + j (a)$ . The remainder identity $j_{p} (a + h) - j_{p} (a) - D j (a) h = j (a + h) - j (a) - D j (a) h$ proves $D j_{p} (a) = D j (a)$ directly from [F1]. In positive source and target dimensions, the first coordinate difference quotients cancel the constant $p$ ; induction on the length of a nonempty word of coordinate derivatives shows that all those derivatives of $j_{p}$ equal those of $j$ . The order-zero map $j_{p}$ is continuous because translation preserves Euclidean distances. Hence $j_{p}$ is $C^{r}$ , including $r = \infty$ . In zero source or target dimension, the stated $C^{r}$ convention and the zero derivative give the same conclusions.

1.2givenF1F2F3

To justify the tangent convention intrinsically, let $k : E \to W$ be another admissible parametrization with $k (b) = j (a) = x$ , first with both parameter dimensions positive. The immersion normal form in [F3] gives a $C^{1}$ map $L$ on an ambient neighbourhood of $x$ satisfying $L (j (v)) = v$ on a domain neighbourhood $V$ of $a$ : project onto the inclusion coordinates and undo the source coordinate change. Shrink $V$ so that $j (V)$ is inside that ambient neighbourhood. Because $j$ is a homeomorphism onto $W$ , $j (V)$ is relatively open in $W$ . By continuity of $k$ and [F2], a neighbourhood of $b$ maps into $j (V)$ . There the transition $ϕ = L \circ k$ is $C^{1}$ , satisfies $ϕ (b) = a$ and $j \circ ϕ = k$ , and the chain rule gives $D k (b) = D j (a) D ϕ (b)$ . Consequently $im D k (b) \subseteq im D j (a)$ .

2.1givenF2step 1.1

The equality $j_{p} (a) = j_{p} (b)$ implies $j (a) = j (b)$ and hence $a = b$ . Translation and its inverse preserve distances, and thus are continuous on Euclidean spaces and their subspaces. The inverse of $j_{p} : D \to W_{p}$ is $x \mapsto j^{- 1} (x - p)$ ; it is continuous by the given homeomorphism and the subspace property in [F2]. Thus $j_{p}$ is a homeomorphism onto $W_{p}$ . Its derivative is injective by step 1.1, so it is an embedded disc with parameter dimension $d$ .

2.2givenF1step 1.2

Interchanging $j$ and $k$ in step 1.2 gives the reverse inclusion, so the tangent subspace is independent of parametrization, even if the parameter dimensions were initially different. If one parameter dimension is zero, the image $W$ is a singleton. An injective parametrization of that image from a nonempty open ball of positive dimension is impossible, since such a ball has two distinct points. Thus the other parameter dimension is also zero, and both derivative images are the zero subspace by [F1]. This also covers ambient dimension zero, since an injective derivative into the zero space forces parameter dimension zero.

3.1givenstep 1.1step 2.1step 1.2step 2.2∎

The explicit basepoint hypotheses give $j_{p} (0) = p + j (0) = p$ . Steps 1.1 and 2.1 show that $W_{p}$ is an embedded $d$ -disc, and steps 1.2 and 2.2 justify applying the tangent convention at each basepoint. Therefore $T_{p} W_{p} = im D j_{p} (0) = im D j (0) = T_{0} W$ . For $d = 0$ , $D = {0}$ , $W = {0}$ , and $W_{p} = {p}$ , with zero tangent subspace as asserted.

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