How do the Christoffel symbols on an abstract manifold relate to those on submanifolds?

Mathematics Asked on January 1, 2022

Let $$(M,g)$$ be a Riemannian manifold of dimension $$N$$ with (Levi-Civita) connection $$nabla$$.

I have seen the following definition of Christoffel symbols: For a given smooth moving frame $$A=(A_1,dots, A_N)$$ (i.e. $$A(p)$$ is a basis for the tangent space $$TM_p$$ at every point $$pin Usubset M$$ where $$U$$ is open), the Christoffel symbols are (locally) defined as the unique $$mathcal C^infty(U)$$-functions $$Gamma_{n,m}^k$$ such that $$nabla_{A_n} A_m =Gamma_{n,m}^k A_k$$ for $$1le m,nle N$$, where I am using the Einstein convention.

I have seen the following definition for sub-manifolds $$Msubset mathbb R^a$$, $$ainmathbb N$$: They are defined (locally) as the unique smooth functions $$Gamma_{n,m}^k$$ such that $$left(frac{partial^2 f}{partial x^n x^m}right)^top=Gamma_{n,m}^k frac{partial f}{partial x^k}$$ for all $$1le n,mle N$$, where $$f:Omegato M$$ is a local parametrization of $$M$$ with an open set $$Omegasubsetmathbb R^N$$ and $$^top$$ denotes the tangential projection onto the tangent space of $$M$$.

Why are these two definitions compatible?

My attempt: Let $$Msubsetmathbb R^a$$ be an $$N$$-dimensional smooth manifold that inherits the Riemannian metric from $$mathbb R^a$$ by restriction. Let $$f=(f^1,f^2,dots,f^N)$$ be a local parametrization around $$pin M$$. Take the local frame $$A_k=frac{partial f}{partial x^k}$$ and let $$nabla^top$$ be the connection that $$M$$ inherits from $$mathbb R^a$$, i.e. if $$X$$ and $$Y$$ are two tangential vector fields to $$M$$, then $$nabla^top_X Yoverset{text{Def.}}=(nabla_{widetilde X}widetilde Y)|_M^top$$, where $$widetilde X$$ and $$widetilde Y$$ are any smooth extensions of $$X,Y$$ to an open subset of $$mathbb R^n$$, and $$nabla$$ is defined by $$nabla_v widetilde Y = v^jfrac{partial widetilde Y^i}{partial x^j} left.frac{partial}{partial x^i}right|_pin Tmathbb R^a_p$$ for any tangent vector $$vin Tmathbb R^a_p$$. I want to show that $$nabla_{A_n} A_m=left(frac{partial^2 f}{partial x^npartial x^m}right)^top$$. For this, fix $$n,min{1,dots, N}$$ and let $$X=A_n=frac{partial f}{partial x^n}, Y=A_m=frac{partial f}{partial x^m}$$. I computed:

$$begin{split} X_p(Y^i)=frac{partial f^j}{partial x^n}(p)cdot frac{partialleft(frac{partial f^i}{partial x^m}right)}{partial x^j}(p)=frac{partial f^j}{partial x^n}(p)cdotfrac{partial^2 f^i}{partial x^mpartial x^j}(p). end{split}$$

But shouldn’t I get $$X_p(Y^i)=frac{partial^2 f^i}{partial x^npartial x^m}(p)$$ for the definitions to work out? Where did I go wrong?

Recall that in your formula, you're writing $$v=sum v^j A_j$$ (with $$A_j = partial f/partial x^j$$), so in the case of the vector field $$v=X=partial f/partial x^n$$, you will have $$v^j = delta^j_n$$.