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:

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).

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?

One Answer

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 $v^j = delta^j_n$.

Answered by Ted Shifrin on January 1, 2022

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