Are differential geometric and physics conventions for covariant derivatives consistent?

Question

In a differential geometric setting, the covariant derivative can be defined as a map $\nabla_X:\Gamma(TM)\to\Gamma(TM)$, for any vector field $X\in\Gamma(TM)$, satisfying certain conditions. In other words, for any vector field, it maps vector fields into other vector fields. This definition is then readily extended to maps between arbitrary tensor fields. Given a local basis $\{\partial_i\}_i$ around some $p\in M$, it can be characterised via the Christoffel symbols as $$\nabla_i\partial_j\equiv \nabla_{\partial_i}\partial_j =\Gamma^k_{~~ij}\partial_k.\tag1$$ Similarly, we get local expressions such as $$\nabla_i Y = \nabla_i (Y^j \partial_j) = (\partial_i Y^j + Y^\ell Y^k \Gamma^j_{~\ell k})\partial_j.\tag2$$

So far, so good. My confusion arises when trying to match this with the notation used in more physical contexts. For example, consider these lecture notes ([29:46] on youtube). Here, they denote the covariant basis as $\vec S_\alpha$, and write the Christoffel symbol as $$\Gamma^\gamma_{~\alpha\beta}=\vec S^\gamma\cdot\partial_\beta \vec S_\alpha.\tag3$$ When writing this, they're assuming to be dealing with embedded surfaces, so taking the standard derivative still makes sense, and I can match this expression with (1) assuming the covariant derivative to be the projection of the standard derivative on the tangent surface.

However, from (3) they derive $$\vec S^\gamma\cdot(\nabla_\alpha\vec S_\beta) = \vec S^\gamma\cdot ( \partial_\alpha\vec S_\beta - \Gamma^\omega_{~\alpha\beta}\vec S_\omega ) = 0.\tag4$$ This appears now to be in direct contrast with (1), as $\vec S_\alpha$ in (4) should correspond to the local basis for the tangent space, $\partial_i$, in (1). In fact, following this notation, $\nabla_\alpha \vec S_\beta$ is normal to the surface, which seems in direct contrast with the definition of covariant derivative in (1), an an object mapping tangent vectors into tangent vectors. So what gives? Why are these two notations seemingly in contrast? Is there a more formal way to understand precisely what kind of object the covariant derivative is in the latter convention?

Answer 1

I have read the same book as the one you linked, and I can say that it does have some technical inaccuracies that slowly build up to the confusion that you are having right now. I will try to list them in order. I will denote the standard covariant basis by $\mathbf{e}_i$, and I will use Greek letters for surface indices and English alphabet for ambient (Euclidean space) indices.

1. There is no such thing as "contravariant basis vectors". What the book refers to as "contravariant basis vectors" are actually basis covectors. In other words, they are one-forms that are linear maps from vectors to scalars. Therefore, expressions like $$\mathbf{e}^i \cdot\mathbf{e}_j = \delta^i_j$$ are incorrect because you can't take the dot product of an object that is not even a vector. Instead, the correct expression should be $$\mathbf{e}^i \left(\mathbf{e}_j\right) = \delta^i_j$$ where the one-form $\mathbf{e}^i$ is acting on the vector $\mathbf{e}_j$. The other expressions should all be modified accordingly.

2. From the very beginning, the book uses an incorrect definition of $\nabla_i\mathbf{e}_j$ in Euclidean space as $$\nabla_i\mathbf{e}_j = \frac{\partial \mathbf{e}_j}{\partial x^i} -\Gamma^k_{ij}\mathbf{e}_k = \mathbf{0}$$ which is a correct evaluation but an incorrect definition. The book treated $\mathbf{e}_j$ as a $(0,1)$ tensor component with a lower index, which is most certainly wrong as $\mathbf{e}_j$ is a vector which is $(1,0)$.
   The correct way to define the covariant derivative for a vector $\mathbf{v}=v^i\mathbf{e}_i$ is $$\nabla_i v^m = (\nabla\mathbf{v})\left(\mathbf{e}_i,\mathbf{e}^m\right) = (\nabla\mathbf{v})^{\;m}_i = \frac{\partial v^m}{\partial x^i}+\Gamma^m_{ik}v^k \\ \nabla_i \mathbf{v} = \left(\frac{\partial v^m}{\partial x^i}+\Gamma^m_{ik}v^k\right)\mathbf{e}_m$$ where it must be noted that the $v^m$ and $v^k$ are the components of $\mathbf{v}$ which are numbers. Therefore, the correct way to compute $\nabla_i\mathbf{e}_j$ is to let $\mathbf{v}=\mathbf{e}_j$ in the above definition. The components are therefore all zero except the $j$-th one which is equal to one. All of them are constant so the first term $\partial v^m/\partial x^i$ is zero. Applying the same logic to the second term, we see that the only non-vanishing term in the sum is the $j$-th one, which gives $\Gamma^m_{ij} (1) = \Gamma^m_{ij}$. Adding in the basis from the second line above, we have $$\nabla_i \left(\mathbf{e}_j\right) = \Gamma^m_{ij}\mathbf{e}_m = \partial_i\mathbf{e}_j$$ which is precisely that of the partial derivative. This is correct because the (Levi-Civita connection) covariant derivative in Euclidean space is simply the partial derivative because its affine nature supplies a canonical parallel transport. It is definitely not zero unlike what the book claims.

3. All of the above leads to the incorrect statement that $\nabla_\alpha \mathbf{e}_\beta$ is normal to the surface. It most certainly is not. The surface covariant derivative $\nabla_\alpha$ is simply the Euclidean partial derivative along the surface coordinate $s^\alpha$, with the normal component removed. The reason for the error is once again because the book used the incorrect definition $$\nabla_\alpha\mathbf{e}_\beta = \frac{\partial \mathbf{e}_\beta}{\partial s^\alpha} -\Gamma^\gamma_{\alpha\beta}\mathbf{e}_\gamma$$ which leads to the incorrect conclusion that $\nabla_\alpha\mathbf{e}_\beta =\mathbf{0}$ on the surface. In general, $\partial_\alpha \mathbf{e}_\beta$ lives in the ambient Euclidean space and therefore has components both normal and tangent to the surface. The normal component is given by the second fundamental form $\mathbb{I}_{\alpha\beta}$ multiplied by the unit normal vector $\hat{\mathbf{n}}$, while the tangential component is the surface covariant derivative $\nabla_\alpha\mathbf{e}_\beta = \Gamma^\gamma_{\alpha\beta}\mathbf{e}_\gamma$, which is exactly analogous to the correct definition I gave above for Euclidean space. This formula also precisely agrees with that on a pseudo-Riemannian manifold (where the ambient Euclidean space is not assumed to exist). In other words, we have $$\partial_\alpha\mathbf{e}_\beta = \Gamma^\gamma_{\alpha\beta}\mathbf{e}_\gamma + \mathbb{I}_{\alpha\beta}\hat{\mathbf{n}} = \nabla_\alpha \mathbf{e}_\beta + \mathbb{I}_{\alpha\beta}\hat{\mathbf{n}}$$

For more information on the first point, see this post. For more information on why the book's definition is wrong, see this post. Lastly, for the correct derivations, see this video which explains it very clearly.