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<p>In this paper, we introduce the notion of developments of curves with respect to symmetric tensors which is a positive codimensional extension of the classical notion of development of curves, and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. The existence part of the fundamental theorem for submanifolds is a direct corollary of our result. Our method also provides a geometric construction of such an isometric immersion when it exists.</p>

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Developments of curves with respect to symmetric tensors and existence of isometric immersions with prescribed second fundamental form

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • Chengjie Yu

摘要

In this paper, we introduce the notion of developments of curves with respect to symmetric tensors which is a positive codimensional extension of the classical notion of development of curves, and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. The existence part of the fundamental theorem for submanifolds is a direct corollary of our result. Our method also provides a geometric construction of such an isometric immersion when it exists.