  This paper is motivated by Davenport's problem and the subsequent work
regarding badly approximable points in submanifolds of a Euclidian space. We
study the problem in the area of twisted Diophantine approximation and present
two different approaches. The first approach shows that, under a certain
restriction, any countable intersection of the sets of weighted badly
approximable points on any non-degenerate C^1 submanifold of R^n has full
dimension. In the second approach we introduce the property of isotropically
winning and show that the sets of weighted badly approximable points are
isotropically winning under the same restriction as above.
