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Aspects of Global Analysis of Circle-Valued Mappings

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Topics in Mathematical Analysis and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 94))

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Abstract

We deal with the minimum number of critical points of circular functions with respect to two different classes of functions. The first one is the whole class of smooth circular functions and, in this case, the minimum number is the so called circular \(\varphi\) -category of the involved manifold. The second class consists of all smooth circular Morse functions, and the minimum number is the so called circular Morse–Smale characteristic of the manifold. The investigations we perform here for the two circular concepts are being studied in relation with their real counterparts. In this respect, we first evaluate the circular \(\varphi\)-category of several particular manifolds. In Sect. 5, of more survey flavor, we deal with the computation of the circular Morse–Smale characteristic of closed surfaces. Section 6 provides an upper bound for the Morse–Smale characteristic in terms of a new characteristic derived from the family of circular Morse functions having both a critical point of index 0 and a critical point of index n. The minimum number of critical points for real or circle valued Morse functions on a closed orientable surface is the minimum characteristic number of suitable embeddings of the surface in \(\mathbb{R}^{3}\) with respect to some involutive distributions. In the last section we obtain a lower and an upper bound for the minimum characteristic number of the embedded closed surfaces in the first Heisenberg group with respect to its noninvolutive horizontal distribution.

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Acknowledgements

The present material was elaborated in the period of the Fall Semester 2013 when the first author was a Mildred Miller Fort Foundation Visiting Scholar at Columbus State University, Georgia, USA. He takes this opportunity to express all the thanks for the nice friendship showed and for the excellent facilities offered during his staying.

The third author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-ID-PCE-2011-3-0994.

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Authors and Affiliations

  1. Department of Mathematics, “Babeş-Bolyai” University, M. Kogălniceanu 1, 400084, Cluj-Napoca, Romania

    Dorin Andrica, Dana Mangra & Cornel Pintea

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  1. Dorin Andrica
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  2. Dana Mangra
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  3. Cornel Pintea
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Correspondence to Dorin Andrica .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics, University of Pécs, Pécs, Hungary

    László Tóth

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Andrica, D., Mangra, D., Pintea, C. (2014). Aspects of Global Analysis of Circle-Valued Mappings. In: Rassias, T., Tóth, L. (eds) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-06554-0_4

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