The Support Vector Machine (SVM) is a powerful tool for classification. We generalize SVM to work with data objects that are naturally understood to be lying on curved manifolds, and not in the usual d-dimensional Euclidean space. Such data arise from medial representations (m-reps) in medical images, Diffusion Tensor-MRI (DT-MRI), diffeomorphisms, etc. Considering such data objects to be embedded in higher dimensional Euclidean space results in invalid projections (on the separating direction) while Kernel Embedding does not provide a natural separating direction. We use geodesic distances, defined on the manifold to formulate our methodology. This approach addresses the important issue of analyzing the change that accompanies the difference between groups by implicitly defining the notions of separating surface and separating direction on the manifold. The methods are applied in shape analysis with target data being m-reps of 3 dimensional medical images.