The paper concerns the problem when a local diffeomorphism \(f:\mathbb R^n\to\mathbb R^n\) is bijective. Let \(g\) be a complete Riemannian metric on \(\mathbb R^n\) and \(h:\mathbb R^n\to\mathbb R\) be a smooth function. Let \(\Delta ^{(g)}h\) be its gradient relative to \(g\) i.e. \(g_x(\Delta ^{(g)}h,w)=dh_x(w)\) for all \(w\in \mathbb R^n\). The main result is: A local diffeomorphism \(f=(f_1,\dots ,f_n):\mathbb R^n\to\mathbb R^n\) is bijective if there exists a complete Riemannian metric \(g\) on \(\mathbb R^n\) such that for every non-zero \(v=(v_1,\dots ,v_n)\in\mathbb R^n\) the smooth vector field \[ \frac{\Delta ^{(g)}f_v(x)}{|\Delta ^{(g)}f_v(x)|^2_g} \] is complete on \(\mathbb R^n\), where \(f_v(x):=\sum_{i=1}^nf_i(x)v_i\). This implies: A local diffeomorphism \(f:\mathbb R^n\to\mathbb R^n\) is bijective if there exists a complete Riemannian metric \(g\) on \(\mathbb R^n\) such that for every non-zero \(v=(v_1,\dots ,v_n)\in\mathbb R^n\) the function \(f_v\) satisfies the Palais-Smale condition i.e. for each sequence \(x_k\in\mathbb R^n\) such that \(f_v(x_k)\) is bounded and \(|df_v(x_k)|\to 0\) there exists a convergent subsequence \({x}_{k_m}\) of \({x}_{k}\). As an application the author obtain new criteria for invertibility of polynomial mappings with constant non-zero jacobian (the jacobian conjecture) in terms of the Łojasiewicz exponent of this mapping at infinity.