Let $(M,g)$ be a Riemannian manifold of dimension $n$ and let $g_{ij}$ be the matrix of the components of $g$ in coordinates. Since $g_{ij}$ is symmetric and positive-definite, it can diagonalized by $g_{ij} = PDP^{T}$, with $D = \text{diag}(\lambda_1, \ldots, \lambda_n)$ diagonal and $P$ orthogonal. For $s \in \mathbb{R}$, define the matrix power $g_{ij}^s := PD^sP^T$, where $D^s = \text{diag}(\lambda_1^s, \ldots, \lambda_n^s)$.
Is there a link between the matrix $g_{ij}^{-1/2}$ and the Christoffel symbols $\Gamma^i_{jk}$? For instance, denoting the matrix by $G := g_{ij}$, if I consider the derivative: \begin{equation} \frac{\partial G^{-1/2}}{\partial x^k} = -\frac{1}{2} G^{-1}\frac{\partial G}{\partial x^k}G^{-1/2}, \end{equation} several "ingredients" of the Christoffel symbols appear. I know that the following holds when the Christoffel are contracted: \begin{equation} \Gamma^i_{ij} = \frac{1}{2}g_{ik}^{-1}\frac{\partial g_{ik}}{\partial x^j}, \end{equation} but I'd be interested to know if a relation involving $g_{ij}^{-1/2}$ exists before contraction. On another note, I derived the expression of the derivative of $G^{-1/2}$ using the usual approach when taking derivatives of matrices (i.e., taking the derivative of the identity $G^{-1/2}GG^{-1/2} = I$), but from I read in this paper https://epubs.siam.org/doi/10.1137/S089547989528274X (Theorem 3.5 with $\alpha = -1/2$), there might be additional terms...
I try to interpret the following "symmetry" relation on the matrix components of a metric: \begin{equation} g_{ti}^{-1/2}\frac{\partial g_{kj}^{-1/2}}{\partial x^t} = g_{tj}^{-1/2}\frac{\partial g_{ki}^{-1/2}}{\partial x^t} \end{equation} Could this be a relation between Christoffel symbols?
For context, I am trying to reinterpret some approaches in numerical geometry in engineering using differential geometry, and I need to reverse-engineer some results to try and make sense of them on manifolds. For the first question, I consider an isometry $F:U \to M$ between $(U \subseteq \mathbb{R}^2, \bar{g})$ and $(M,g)$, where $\bar{g}$ is the Euclidean metric. If $y^j$ are coordinates on $U$, it is my understanding that in coordinates, the components of the differential map $dF$ satisfy: \begin{equation} \frac{\partial F^i}{\partial y^j} = g_{ik}^{-1/2}O_{kj} \end{equation} for some orthogonal matrix $O$, since $F$ pulls back the metric $g$ to the Euclidean metric, that is: \begin{equation} \bar{g}_{ij} = \delta_{ij} = \left(\frac{\partial F^k}{\partial y^i}\right)^T g_{kl} \frac{\partial F^l}{\partial y^j}, \end{equation} which is satisfied if the Jacobian matrix has the form written above. Thus, the square root matrix of 1. originates from this.
The symmetry relation of 2. is obtained as follows: I consider a map $F:U \to M$ that is not necessarily an isometry or even smooth, and I suppose that I can write its Jacobian matrix in coordinates as: \begin{equation} \frac{\partial F^i}{\partial y^j} = g_{ik}^{-1/2}R_{kj}, \end{equation} with $R$ a rotation matrix. Then I consider what kind of constraints there should be on the components of the metric to have at least a $C^2$ map. The symmetry relation is obtained by imposing that the second derivatives of $F^i$ are equal, to ensure $C^2$ continuity.
Please feel free to correct any nonsense in what I said, I am learning DG on my own using Lee's Introduction to smooth and Riemannian manifolds. Thanks for your help!