$S_1 ,S_2$ submanifolds of $\mathbb{T}^2$. Does there exists a translation $L$, such that $T(S_1)\pitchfork S_2$?

Question

Let $\mathbb{T}^2 = [0,1]^2/\sim$ be the torus, and $S_1,S_2 \subset \mathbb{T}^2$ are smooth $1$-manifolds such that $$S_i =\dot{\bigcup}_{j=1}^{n_i} C_{j}^{i}, $$ where each $C_j^{i}$ is a $1$-manifold diffeomorphic to $\mathbb{S}^1$.

Let $L_{\varepsilon,\delta}: \mathbb{T}^2 \to \mathbb{T}^2 $ be the smooth diffeomorphism $$L_{\varepsilon,\delta}\left([x,y]\right) = [x+\varepsilon,y+\delta].$$

My Question: Is it possible guarantee that there exists $(\varepsilon, \delta)\in \mathbb{R}^2$, such that $L_{\varepsilon,\delta}(S_1) \pitchfork S_2$?

I don't have any good idea on how to solve this problem, can anyone help me?

I tried to find a smart way to apply the transversality theorem but it led me to nowhere.

Answer 1

Just for the records, the definitions that I'm using:

Definition 1: Let $M,N$ be smooth manifolds, $S \subset N$ a smooth submanifold and $f: M \to N$ a smooth funtion. Then $f \pitchfork S$ if for every $p$ $\in$ $M$ occurs one of the following conditions:

• $f(p) \not \in S$
• $f(p) \in S$ and $\text{d}f_p (T_{p}M) + T_{f(p)}S = T_p N. $

Definition 2: Let $M$ be a smooth manifold and $S_1, S_2 \subset M$ smooth submanifolds. Then $S_1 \pitchfork S_2$ if for every $p$ $\in$ $S_1 \cap S_2$ follows $T_p S_1 + T_p S_2 = T_p M$.

Besides that, there exists the following well-known theorem

Theorem: Let $M, N, S$ be smooth manifolds, and $F: M\times N \to S$ a smooth function such that $F \pitchfork Z$, where $Z$ is a smooth submanifold of $S$. Then $\{p \in N;$ $F(\cdot,p) \pitchfork S\}$ is a residual set.

The above theorem also holds if we change word "residual" to "full measure set" (complementary of a null set).

The demonstration follows immediately from the fact that the function

$$F: S_1 \times \mathbb{R}^2 \to \mathbb{T}^2 $$ $$([p],\varepsilon,\delta)\mapsto [p + (\varepsilon,\delta)] $$

satisfies $F \pitchfork S_2$, because $\text{d}F_q$ is obviously surjective. Then follows from Theorem 1 that the set $\{(\varepsilon,\delta) \in \mathbb{R}^2;$ $F(\cdot,\varepsilon,\delta) \pitchfork S_2\}$ isn't empty.