In many cases of high dimensional data analysis, data points may lie on manifolds of very complex shapes/geometries. Thus, the usual Euclidean distance may lead to suboptimal results when utilized in clustering or visualization operations. In this work, we introduce a new distance definition in multi-dimensional spaces that preserves the topology of the data point manifold. The parameters of the proposed distance are discussed and their physical meaning is explored through 2 and 3-dimensional synthetic datasets. A robust method for the parameterization of the algorithm is suggested. Finally, a modification of the well-known k-means clustering algorithm is introduced, to exploit the benefits of the proposed distance metric for data clustering. Comparative results including other established clustering algorithms are presented in terms of cluster purity and V-measure, for a number of well-known datasets.