A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds

Given a weight vector $\tau=(\tau_{1}, \dots, \tau_{n}) \in \mathbb{R}^{n}_{+}$ with each $\tau_{i}$ bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set of $\tau$-approximable points points over a manifold $\mathcal{M}$, where $\mathcal{M}$ is twice continuously differentiable. From this we produce a lower bound for the set of $\psi$-approximable points over a manifold where $\psi$ is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in arXiv:1712.03761, but we use an alternative mass transference style theorem.