Let M X ( r , ξ ) M_X(r,\xi ) be the moduli space of stable vector bundles, on a smooth complex projective curve X X with genus ( X ) ≥ 3 \text {genus}(X)\, \geq \, 3 , of rank r r and fixed determinant ξ \xi such that deg ⁡ ( ξ ) \deg (\xi ) is coprime to r r . If E E is a vector bundle M X ( r , ξ ) M_X(r,\xi ) whose restriction to every Hecke curve in M X ( r , ξ ) M_X(r,\xi ) is trivial, we prove that E E is trivial.