Despite the great performance of deep learning models in many areas, they still make mistakes and underperform on certain subsets of data, i.e. error slices. Given a trained model, it is important to identify its semantically coherent error slices that are easy to interpret, which is referred to as the error slice discovery problem. However, there is no proper metric of slice coherence without relying on extra information like predefined slice labels. Current evaluation of slice coherence requires access to predefined slices formulated by metadata like attributes or subclasses. Its validity heavily relies on the quality and abundance of metadata, where some possible patterns could be ignored. Besides, current algorithms cannot directly incorporate the constraint of coherence into their optimization objective due to absence of an explicit coherence metric, which could potentially hinder their effectiveness. In this paper, we propose manifold compactness, a coherence metric without reliance on extra information by incorporating the data geometry property into its design, and experiments on typical datasets empirically validate the rationality of the metric. Then we develop Manifold Compactness based error Slice Discovery (MCSD), a novel algorithm that directly treats risk and coherence as the optimization objective, and is flexible to be applied to models of various tasks. Extensive experiments on the benchmark and case studies on other typical datasets demonstrate the superiority of MCSD.