# What is the standard smooth structure on a regular submanifold?

A subset $$S \subseteq M$$ is a regular submanifold of dimension $$k$$ of a smooth manifold $$M$$ of dimension $$n$$ if for every point $$p \in S$$, there exists a chart $$(U_p, \varphi_p)$$ around $$p$$ in the smooth structure of $$M$$ such that the intersection $$U_p \cap S = \varphi_p'(U_p)$$ where $$\varphi_p'$$ some function which may be obtained by sending $$n - k$$ of the coordinate functions of $$\varphi_p$$ to zero.

The above is supposed to be a definition, but I don't understand how it defines the topology or the smooth structure. Is the subspace topology implicit in the above definition? Or is it supposed to be another assumption?

Let $$\varphi_S: U_p \cap S \to \mathbb{R}^k$$. How can we show $$(U_p \cap S, \varphi_S)$$ is a chart on $$S$$? Does this process of constructing charts uniquely determine the smooth structure?

• That those $\varphi_S$ forms a chart on $S$ is a direct checking. Did you check that? May 11, 2021 at 4:55
• I don't quite see it. Are the adapted charts also smoothly compatible charts on $M$? I don't see how to prove that the charts $\varphi_S$ are smoothly compatible with each other.
– lanf
May 11, 2021 at 14:15
• Given any two $\varphi_p, \varphi_q$ you obtain two maps $\varphi^S_p : U_p \cap S \to \mathbb R^k$ and similar for $\varphi^S_q$. Try to write $(\varphi^S_q)^{-1} \circ \varphi^S_p$ in terms of $\varphi_q^{-1}\circ \varphi_p$. May 11, 2021 at 16:35

Yes, $$S$$ is given the subspace topology inherited from $$M$$. But the phrase
there exists a chart $$(U_p, \varphi_p)$$ around $$p$$ in the smooth structure of $$M$$ such that the intersection $$U_p \cap S = \varphi_p'(U_p)$$ where $$\varphi_p'$$ is some function which may be obtained by sending $$n - k$$ of the coordinate functions of $$\varphi_p$$ to zero
does not make much sense. The chart $$\varphi_p : U_p \to V_p$$ is a homeomorphism onto an open $$V_p \subset \mathbb R^n$$. Now the set $$U_p$$ occurs on both sides of the equation $$U_p \cap S = \varphi_p'(U_p)$$ and therefore $$\varphi_p'$$ must be a map on $$U_p$$ and taking values in $$M$$ - but this has no coordinate functions which can take the value $$0$$. What is meant is that $$\varphi_p(U_p \cap S) = V_p \cap \mathbb R^n_k$$, where $$\mathbb R^n_k = \{(x_1,\dots,x_n) \in \mathbb R^n \mid x_{k+1} = \dots = x_n = 0\}$$. Then you can define $$\varphi'_p= \pi^n_k \circ \varphi_p \mid_{U_p \cap S}: U'_p = U_p \cap S \to V'_p$$ where $$\pi^n_k$$ denotes projection onto the first $$k$$ coordinates and $$V'_p = \pi^ n _k(V_p \cap \mathbb R^n_k)$$.
Then you can prove that the $$\varphi'_p$$ form an atlas on $$S$$ which has smooth transition functions and thus determines a smooth structure on $$S$$.