Let $(M,G)$ be a Riemannian metric and $V$ a $1$-form on $M$. Is there a coordinates-free, geometrical interpretation of the scalar $$G^{ij} \partial_i V_j$$ with summation over repeated indices understood?
In particular, let $\iota: S \hookrightarrow M $ be a submanifold of $M$, $g$ be the metric on $S$ induced by the ambient space metric, and $v$ be the pull-back of $V$ along the inclusion (hence a $1$-form on $S$).
Is the following true (for any choice of charts)? $$\sum_{i,j = 1}^{\dim{S}} g^{ij}\partial_i v_j = \sum_{h,k = 1}^{\dim{M}} G^{hk}\partial_h V_k $$
Context This quantity appears computing the co-differential $\delta: \Omega^k \to \Omega^{k-1}$ of a $1$-form $V = V_i \, dx^i$ :$$\delta V = V_i \, \delta dx^i - G^{ij} \partial_i V_j$$ with $\Omega^k$ denoting the space of $k$-forms on a manifold; see e.g. Jürgen Jost, Riemannian Geometry and Geometric Analysis.