  In this paper, we study the weighted $n$-dimensional badly approximable
points on manifolds. Given a $C^n$ differentiable non-degenerate submanifold
$\mathcal{U} \subset \mathbb{R}^n$, we will show that any countable
intersection of the sets of the weighted badly approximable points on
$\mathcal{U}$ has full Hausdorff dimension. This strengthens a result of
Beresnevich by removing the condition on weights and weakening the smoothness
condition on manifolds. Compared to the work of Beresnevich, our approach
relies on homogeneous dynamics. It turns out that in order to solve this
problem, it is crucial to study the distribution of long pieces of unipotent
orbits in homogeneous spaces. The proof relies on the linearization technique
and representations of $\mathrm{SL}(n+1,\mathbb{R})$.
