Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

Chen Yun Lin, Hau Tieng Wu

Research output: Contribution to journalArticlepeer-review


We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in manifold learning. In this paper, we provide a positive answer to the problem. Specifically, we use eigenvector fields to construct local coordinate charts with low distortion, and show that the distortion constants depend only on geometric properties of manifolds with metrics in the little Hölder space c2 , α. Next, we use the coordinate charts to embed the entire manifold into a finite dimensional Euclidean space. The proof of the results relies on solving the elliptic system and providing estimates for eigenvector fields and the heat kernel and their gradients. We also provide approximation results for eigenvector field under the c2 , α perturbation.

Original languageEnglish (US)
Article number126
JournalCalculus of Variations and Partial Differential Equations
Issue number5
StatePublished - Oct 1 2018

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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