Quantum states defined over a parameter space form a Grassmann manifold. To capture the geometry of the associated gauge structure, gauge-invariant quantities are essential. We employ the projector of a multilevel system to quantify the quantum distance between states. Using the multidimensional scaling method, we transform the quantum distance into a reconstructed manifold embedded in Euclidean space. This approach is demonstrated with examples of topological systems, showcasing their topological features within these manifolds. Our method provides a comprehensive view of the manifold rather than focusing on local properties.

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• Electronic structure
• Topological insulators
• Machine learning