When dealing with pattern recognition problems one encounters different types of prior knowledge. It is important to incorporate such knowledge into the classification method at hand. A common prior knowledge is that many datasets are on some kinds of manifolds. Distance-based classification methods can make use of this by a modified distance measure called geodesic distance. We introduce a new kind of kernels for a support vector machine (SVM) which incorporates geodesic distance and therefore is applicable in cases where such transformation invariance is known. Experiments results show that the performance of our method is comparable to that of other state-of-the-art methods, such as SVM-based Euclidean distance.