# On the Morse Index for Geodesic Lines on Smooth Surfaces Embedded in ℝ3

Article

The paper is devoted to the calculation of the Morse index on geodesic lines upon smooth surfaces embedded into the 3D Euclidean space. The interest in this theme is created by the fact that the wave field composed of the surface waves slides along the boundaries guided by the geodesic lines, which, generally speaking, give birth to numerous caustics. The same circumstance takes place in problems of the short-wave diffraction by 3D bodies in the shadowed part of the surface of the body, where the creeping waves arise. Two types of geodesic flows are considered upon the surface when they are generated by a point source and by an initial wave front, for instance, by the light-shadow boundary in the short-wave diffraction by a smooth convex body. The position of the points where geodesic lines meet caustics, i.e., focal points, is found and it is proved that all focal points are simple (not multiple) irrespective of the geometric structure of the caustics arisen. The mathematical techniques in use are based on the complexification of the geometrical spreading problem for a geodesics/rays tube.

## References

1. 1.
J. W. Milnor, Morse Theory, Princeton University Press, Princeton, New Jersey (1963).
2. 2.
V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Springer (1981).Google Scholar
3. 3.
V. I. Arnold, “Characteristic class entering in quantization conditions,” Func. Analysis Appl., 1, 1–13 (1967).
4. 4.
M. M. Popov, “On calculating the Morse index and continuation of ray formulas beyond caustics,” J. Math. Sci., 224, 150–156 (2017).
5. 5.
V. I. Smirnov, A Course in Higher Mathematics, Vol. 4, Pergamon Press, Oxford (1964).
6. 6.
M. M. Popov, Ray Theory and Gaussian Beam Method for Geophysicists, EDUFBA, Salvador-Bahia (2002).Google Scholar
7. 7.
M. M. Popov, “On a method of calculation of geometrical spreading in an inhomogeneous medium with interfaces,” Dokl. Akad. Nauk SSSR, 237, 1059–1062 (1977).Google Scholar
8. 8.
V. I. Smirnov, A Course in Higher Mathematics, Vol. 2, Pergamon Press, Oxford (1964).
{"ft_lang_label":"__label__en","ft_lang_prob":0.80703753,"math_prob":0.9537806,"size":2296,"snap":"2019-43-2019-47","text_gpt3_token_len":580,"char_repetition_ratio":0.118673645,"word_repetition_ratio":0.034883723,"special_character_ratio":0.23867595,"punctuation_ratio":0.19912472,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9600129,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-19T00:56:03Z\",\"WARC-Record-ID\":\"<urn:uuid:fc3401cb-b4a8-4030-8f0a-c4c9e4d607ad>\",\"Content-Length\":\"68854\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f36def26-6858-4ec7-a1c8-12935f5ce3ec>\",\"WARC-Concurrent-To\":\"<urn:uuid:d26de37c-83b1-4f3d-bc78-6a50229ea001>\",\"WARC-IP-Address\":\"151.101.248.95\",\"WARC-Target-URI\":\"https://rd.springer.com/article/10.1007%2Fs10958-019-04577-3\",\"WARC-Payload-Digest\":\"sha1:5PJKGNXZHMWEWX2Y2ENMEBRIHY5LTJTA\",\"WARC-Block-Digest\":\"sha1:RE3E6ZKYBJ4VZXCYRER3P764SNLKYWVE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496669868.3_warc_CC-MAIN-20191118232526-20191119020526-00242.warc.gz\"}"}