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<p>In the 1990s, C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space <i>G</i>/<i>K</i>. This map is called the parallel transport map over <i>G</i>/<i>K</i>. Later, N. Koike extended their theory to the case that <i>G</i>/<i>K</i> is a Riemannian symmetric space of non-compact type. In this paper, more generally, we define the parallel transport map over an affine symmetric space and show that it is an affine submersion with horizontal distribution in the sense of Abe and Hasegawa. Based on this result, we prove the Fredholm property of affine immersions into a Hilbertable space lifted by the parallel transport map. Furthermore, we greatly extend the author’s previous result on weakly reflective submanifolds from the case of compact Riemannian symmetric spaces to the case of affine symmetric spaces.</p>

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The Parallel Transport Map Over Affine Symmetric Space

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  • 出版模式：

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  • 发表日期：

    2025年9月8日

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  • Masahiro Morimoto

摘要

In the 1990s, C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space G/K. This map is called the parallel transport map over G/K. Later, N. Koike extended their theory to the case that G/K is a Riemannian symmetric space of non-compact type. In this paper, more generally, we define the parallel transport map over an affine symmetric space and show that it is an affine submersion with horizontal distribution in the sense of Abe and Hasegawa. Based on this result, we prove the Fredholm property of affine immersions into a Hilbertable space lifted by the parallel transport map. Furthermore, we greatly extend the author’s previous result on weakly reflective submanifolds from the case of compact Riemannian symmetric spaces to the case of affine symmetric spaces.