In machine learning, a high dimensional dataset such as the digital image of human face is often considered as a point set distributed on a (differentiable) manifold. In many cases the intrinsic dimension of this manifold is highly lower than the representation dimension. In order to ease data processing, manifold-based dimension reduction algorithms were put forward since this century. The real purpose of manifold learning (MAL) is to learn a suitable metric in the low dimensional space. One main limitation of the existing MAL algorithms is that they all do not consider the intrinsic curvature of the embedded manifold, which means that the intrinsic geodesic distance cannot be uncovered by these algorithms. The intrinsic geodesic distance on the manifold is measured by Riemannian metric which is highly affected by Riemannian curvature. With this idea in mind, our work proposes to formulate a new algorithm by adding the curvature information to metric learning. We study a new curvature flow from input dataset. By employing this curvature flow, we obtain a Mahalanobis metric which can better uncover the intrinsic structure of the embedded manifold. To show the effectiveness of our proposed method, we compare our algorithm with several traditional MAL algorithms.