# Derivative of basis vector in terms of Christoffel symbols

I would like to derive the formula

$$\partial_{c}\vec{e}^{\,a}=-\Gamma_{bc}^{a}\vec{e}^{\,b}$$

where $$\vec{e}_{a}$$ are the basis vectors on a manifold.

In the lecture, we did it in the following way:

$$0=\partial_{c}(\delta_{b}^{a})=\partial_{c}(\vec{e}^{\,a}\cdot\vec{e}_{\,b})=\vec{e}_{\,b}\cdot\partial_{c}\vec{e}^{\,a}+\underbrace{\vec{e}^{\,a}\cdot\partial_{c}\vec{e}_{\,b}}_{=\Gamma_{bc}^{a}}$$

and therefore

$$\vec{e}_{\,b}\cdot\partial_{c}\vec{e}^{\,a}=-\Gamma_{bc}^{a}$$

Up to here I can follow. Then it is stated: "Multipliying with $$\vec{e}^{\,b}$$ yields the result", but I can't unterstand how.

If I multiply by $$\vec{e}^{\,b}$$ and then sum over $$b$$, we get

$$\vec{e}^{\,b}\cdot (\vec{e}_{\,b}\cdot\partial_{c}\vec{e}^{\,a})=-\Gamma_{bc}^{a}\vec{e}^{\,b}$$

How can we simplify the LHS?

The usual definition of the Christoffel symbols is $$\nabla_\mu {\bf e}_\nu = {\bf e}_\sigma {\Gamma^\sigma}_{\nu\mu}$$ where $$\nabla_\mu$$ is a covaraint derivative and the $${\bf e}_\mu$$ are basis vectors of the tangent space $$T(M)$$. Your formula differs in sign for some reason. Where does it come from? Are the $${\bf e}_a$$ an orthonormal basis, and if so what are the $${\bf e}^a$$? Are they a coframe? If so, the sign is right, but they are not the basis vetors of the tansgent space, but rather of its dual $$T^*(M)$$. Also if this is the case then for a vector $$X= X^a {\bf e}_a$$ we us the fact that evalauting a covector on a vector returns its components, i.e. $${\bf e}^a(X)=X^a$$, to see that $${\bf e}_a {\bf e}^a(X)= {\bf e}_a X^\mu=X.$$ Thus $$\sum_a {\bf e}_a {\bf e}^a$$ is the identity map from $$T(M)\to T(M)$$. No "$$\cdot$$" is needed between the $${\bf e}_a$$ and the $${\bf e}^a$$.
• @ggcg Yes. That's why I said the sign was correct if he was using a basis for $T^*(M)$ rather than for $T(M)$. It was not clear what the ${\bf e}^a$ were. I usually use the notation ${\bf e}^{*a}$ for the dual basis so as to avoid the ambiguity of whether a number $e_1^3$ is the third component of the vctor ${\bf e}_1= e_a^\mu \partial_\mu$ or the first component of the covector ${\bf e}^{*3}= e^{*3}_\mu dx^\mu$. Commented Feb 4, 2021 at 14:00