Aspects of Global Analysis of Circle-Valued Mappings

Dorin Andrica (), Dana Mangra () and Cornel Pintea ()
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Dorin Andrica: “Babeş-Bolyai” University
Dana Mangra: “Babeş-Bolyai” University
Cornel Pintea: “Babeş-Bolyai” University

A chapter in Topics in Mathematical Analysis and Applications, 2014, pp 79-109 from Springer

Abstract: 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 φ $$\varphi$$ -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 φ $$\varphi$$ -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 ℝ 3 $$\mathbb{R}^{3}$$ 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.

Keywords: Circular function; φ $$\varphi$$ -Category; Circular φ $$\varphi$$ -category; Covering map; Fundamental group; Compact surface; Projective space; Morse function; Morse–Smale characteristic; Circular Morse–Smale characteristic; Riemann surface; Poincaré–Hopf Theorem; The first Heisenberg group; Characteristic point; 58E05; (57R70; 57R35; 57M10; 55M30) (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-06554-0_4

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