Density peaks clustering algorithm uses Euclidean distance as a measure of similarity between the samples and it can achieve a good clustering effect when processing the manifold datasets. Utilising this feature, we propose a density peaks clustering algorithm based on geodetic distance and dynamic neighbourhood. This new algorithm measures the similarity between the samples by using geodetic distance, and the number of neighbours K is dynamically adjusted according to the spatial distribution of samples for geodetic distance computation. By choosing geodetic distance as the similarity measure, the problems of manifold dataset clustering can be easily solved, and the clustering is made more effective when the sparse clusters and dense clusters co-exist. The new algorithm was then compared against the other five clustering algorithms on six synthetic datasets and ten real-world datasets. The experiments showed that the proposed algorithm not only outperformed the other conventional algorithms on manifold datasets, but also achieved a very good clustering effect on multi-scale, cluttered and intertwined datasets.