Research Article
Open Access
Proof of Morse’s Lemma
1
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
3
School of Science, Qiqihar University, Qiqihar 161006, China
4
Department of Science, Changchun University of Science and Technology, Changchun 130022, China
Corresponding Author:
Wenliang Gan,
[email protected]
Volume 2, Issue 1
2026
Received: 28 January 2026
Accepted: 09 March 2026
Published: 16 March 2026
Article Information
Volume/Issue
Volume 2, Issue 1, 2026
Pages
54-58
Abstract
In this paper, we present an important property of smooth function germs of orbit tangent space, and apply it to discuss the proof of Morse's Lemma.
Keywords
function germ
Jacobian ideal
diffeomorphism
Morse germ
Morse's Lemma
Data Availability Statement
Data will be made available on request.
Funding
This work was supported in part by the Scientific Research Foundation for the Talent Introduction of Guizhou University of Finance and Economics in 2019 under Grant 2019YJ008; in part by the Fundamental Research Funds in Heilongjiang Provincial Universities under Grant 145309626; in part by the Science Technology Development Plan Project of Jilin Province, China under Grant 20230101186JC; in part by the National Natural Science Foundation of China under Grant 11501051.
Conflicts of Interest
The authors declare no conflicts of interest.
AI Use Statement
The authors declare that no generative AI was used in the preparation of this manuscript.
Ethical Approval and Consent to Participate
Not applicable.
References
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Cite This Article
APA Style
Gan, W., Xiong, Z., Li, Q., & Gao, R. (2026). Proof of Morse’s Lemma. Journal of Mathematics and Interdisciplinary Applications, 2(1), 54–58. https://doi.org/10.62762/JMIA.2026.947879
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TY - JOUR AU - Gan, Wenliang AU - Xiong, Zonghong AU - Li, Qiang AU - Gao, Ruimei PY - 2026 DA - 2026/03/16 TI - Proof of Morse’s Lemma JO - Journal of Mathematics and Interdisciplinary Applications T2 - Journal of Mathematics and Interdisciplinary Applications JF - Journal of Mathematics and Interdisciplinary Applications VL - 2 IS - 1 SP - 54 EP - 58 DO - 10.62762/JMIA.2026.947879 UR - https://www.icck.org/article/abs/JMIA.2026.947879 KW - function germ KW - Jacobian ideal KW - diffeomorphism KW - Morse germ KW - Morse's Lemma AB - In this paper, we present an important property of smooth function germs of orbit tangent space, and apply it to discuss the proof of Morse's Lemma. SN - 3070-393X PB - Institute of Central Computation and Knowledge LA - English ER -
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@article{Gan2026Proof,
author = {Wenliang Gan and Zonghong Xiong and Qiang Li and Ruimei Gao},
title = {Proof of Morse’s Lemma},
journal = {Journal of Mathematics and Interdisciplinary Applications},
year = {2026},
volume = {2},
number = {1},
pages = {54-58},
doi = {10.62762/JMIA.2026.947879},
url = {https://www.icck.org/article/abs/JMIA.2026.947879},
abstract = {In this paper, we present an important property of smooth function germs of orbit tangent space, and apply it to discuss the proof of Morse's Lemma.},
keywords = {function germ, Jacobian ideal, diffeomorphism, Morse germ, Morse's Lemma},
issn = {3070-393X},
publisher = {Institute of Central Computation and Knowledge}
}
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Copyright © 2026 by the Author(s). Published by Institute of Central Computation and Knowledge. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
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