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I am a beginner in differential geometry and I am reading chapter 1 of Differential Geometry of Loring Tu. For a smooth manifold $M$, a Riemannian metric on $M$ is an assignment that assigns $p\in M$ to an inner product on $T_pM$, such that for any smooth vector fields $X,Y$ on $M$, the map $p\mapsto \langle X_p,Y_p \rangle$ is a smooth function on $M$.

Let $(M,\langle,\rangle_M)$ be a Riemannian manifold and $N$ be a submanifold. Then for each $p\in N$, $T_pN$ is a subspace of $T_pM$, so we can naturally define a Riemannian metric on $N$ by letting $\langle v,w\rangle_N=\langle v,w\rangle_M$ for $v,w\in T_pN, p\in N$. But how can we show that this Riemannian metric on $N$ satisfy the smoothness condition? I.e., for any smooth vector fields $X,Y$ on $N$, how can we show the map $p\mapsto \langle X_p,Y_p \rangle$ is a smooth function on $N$?

user302934
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1 Answers1

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Smoothness is a local property, so consider an open neighborhood $U\subseteq M$ of $p \in M$, and extend $X$ and $Y$ to vector fields $\widetilde{X}$ and $\widetilde{Y}$ on $U$. So the function $U \cap N \ni q \mapsto \langle X_q,Y_q\rangle_N$ is the restriction of the smooth map $U \ni q \mapsto \langle \widetilde{X}_q,\widetilde{Y}_q\rangle_M$, which by assumption is smooth. Restrictions of smooth maps are smooth, and so you are done.

If you're a more coordinate-oriented person, you can take coordinates $(x^j, y^\mu)$ adapted to $N$, i.e., such that $N$ is described by $y^\mu = 0$, so that the tangent spaces to $N$ are spanned by the first coordinate fields $\partial_j$. Then the metric matrix has the block form $$\begin{pmatrix} (g_{jk}) & (g_{j\lambda}) \\ (g_{\lambda k}) & (g_{\mu\nu})\end{pmatrix}$$where all the entries are smooth. In particular the first block $(g_{jk})$ is smooth, so you're done.

Ivo Terek
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  • I like your first proof. Thanks. I have one question. Can a vector field on $N$ can be always extended to a vector field on a neighborhood in $M$ of any point in $N$? – user302934 Jul 25 '20 at 06:09
  • I found my answer in https://math.stackexchange.com/questions/119415/extending-a-vector-field-defined-on-a-closed-submanifold – user302934 Jul 25 '20 at 06:14
  • Yes, we actually used that. Since $N$ is a submanifold, take slice coordinates $(x^j,y^\mu)$ as above. You have that $X = \sum X^j\partial_j$ since the $\partial_j$'s span the tangent spaces to $N$. Extend using this same expression (the extension depends on the choice of coordinates) – Ivo Terek Jul 25 '20 at 06:15