We define a new local metric for labeled data, or features, which elongates the standard Euclidean distance in the direction of the label gradient, and explore its utility for supervised dimension reduction (SDR). In particular, we create supervised low-dimensional feature representations by embedding a gradient elongated geodesic distance. Experiments indicate that this procedure leads to faithful visualization of the underlying structure of the feature-label relationship, i.e., the active manifold, when other SDR methods fail. Furthermore, we propose a new method for prediction on unlabeled points, which extends Laplacian learning to leverage the gradient elongated geometry. Extensive numerical comparisons with classic and neural network-based methods indicate that such an approach outperforms other prediction methods when the number of labeled points is small. Finally, we validate the proposed methodology on several real-world datasets, including data on acute respiratory syndrome-coronavirus-2 and data on human stem cell differentiation.