I am currently reading mathematical gauge theory by Hamilton and I am trying to solve problem 1.9.10 which states:
Find and explicit embding:
$$O(n)\hookrightarrow SO(n+1)$$
Now I understand that a submanifold $N\subset M$ is an embedded submanifold if the inclusion map $ i: N\hookrightarrow M$ is a smooth embedding. However, I have never actually worked with embedded subamnifolds using an inclusion map, I have almost always defined them as level sets of some $F:M\rightarrow K$ for some $k$ dimensional smooth manifold. I also don't really see how $O(n)$ is even a submanifold of $SO(n+1)$, since wouldn't $O(n)$ still have determent $\pm 1$ and thus not be in $SO(n_1)$?
Any help or direction would be appreciated.
One needs to come up with a map here, then verify it is an embedding. Fix a basis for $\mathbb{R}^n$. The map $$F: A \mapsto \begin{bmatrix} \det(A) & 0 \\ 0 & A\end{bmatrix}$$ is an injective smooth immersion $O(n) \to SO(n+1)$. As $O(n)$ is compact, all injective smooth immersions $O(n) \to N$, where $N$ is a smooth manifold with or without boundary, are embeddings (this useful fact can be found in chapter 4 of Lee's Introduction to Smooth Manifolds).