Isometric embeddings via heat kernel

2013/05/24 by Xiaowei Wang, Ke Zhu, Wang, Xiaowei +1
Mathematics · #53C42 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Spectral Theory (math.SP) #advanced mathematical theories #math.DG #math.SP #msc:53C42 #msc:58J50

paper · pdf · doi:10.48550/arxiv.1305.5613

39 pages. We remove the Einstein condition in the previous version. The isometric embedding is extended to all compact Riemannian manifolds with smooth metrics. The embedding is improved to C^k regularity for any given k. Proofs are simplified, and less relevant results are removed

openalex publication_date 2013/05/24 · arxiv created 2013/12/05 · arxiv updated 2013/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_t: M->R^q(t), with t>0 sufficiently small and q(t)>>t^-n/2. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As t->0, asymptotic geometry of the embedded images is discussed.

Citations

• Embedding Riemannian Manifolds by the Heat Kernel of the Connection Laplacian
• Random Morse functions and spectral geometry
• The spectral function of an elliptic operator
• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Related

• Conformal Embeddings via Heat Kernel
• Manifold embeddings by heat kernels of connection Laplacian
• Embeddings of Riemannian manifolds with heat kernels and eigenfunctions
• Stability analysis of isometric embeddings
• High-jet relations of the heat kernel embedding map and applications