擴散映射（Diffusion maps）由 Coifman 和 Lafon 所提出，透過將熱核以高斯核取代，以近似流形上的熱擴散現象。本篇論文考慮的情況為流形是嵌入於三維歐幾里得空間中的曲面。我們的目標是分析高斯核的漸進展開，並與熱核的結果進行比較。此外，我們也透過直接計算，考察曲面為單位球面的情形，以驗證該結果與一般情況是否一致。

Diffusion maps, introduced by Coifman and Lafon, approximate heat diffusion on the manifold by replacing the heat kernel with a Gaussian kernel. In this thesis, we consider the specific setting where the manifold is a surface embedded in three-dimensional Euclidean space. Our goal is to analyze the asymptotic expansion of the Gaussian kernel and to compare the result with that of the heat kernel. Furthermore, we examine the case where the surface is the unit sphere through direct calculation to verify that the result agrees with the general case.

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