Gradient of a function restricted to a submanifold
you need a metric: $$ g(\operatorname{grad}f,v)=v(f) \qquad g|_M(\operatorname{grad}f|_M,v_m)=v_m(f|_M) \\ $$ for all tangent vectors (this is the definition), and if $\operatorname{grad}f = V_o+V_t$ is the orthogonal decomposition in the ambient tangent space $$ g|_M(\operatorname{grad}f|_M,v_m)=v_m(f|_M)=v_m(f)=g(\operatorname{grad}f,v_m) =g(V_o+V_t,v_m)=g(V_o,v_m)+g(V_t,V_m)=g(V_t,v_m) $$ so $$ g(\operatorname{grad}f|_M,-)=g(V_t,-) $$ which means by definition that they are equal.