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JOURNALS // Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika // Archive

Izv. Vyssh. Uchebn. Zaved. Mat., 2010 Number 5, Pages 18–24 (Mi ivm6732)

Realizability of the $H_k$-distance functions by homology classes of path spaces

Yu. V. Ershov, E. I. Yakovlev

Nizhni Novgorod State University, Nizhni Novgorod, Russia

Abstract: In the previous papers we constructed and studied mappings $d_k\colon M\times M\to\mathbb R$; we called them the $H_k$-distance functions. The main result of this paper is a theorem about the realizability of generalized distances $d_k(v,w)$, $v,w\in M$, considered as critical values of the length functional $\mathcal L\colon\Omega(M,v,w)\to\mathbb R$ generated by some nontrivial homology classes of the space $\Omega(M,v,w)$ of paths between points $v$ and $w$.

Keywords: Riemannian manifold, path space, distance functions, multivalued functional, extremal.

UDC: 514.764+515.165

Received: 31.03.2008


 English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2010, 54:5, 15–20

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© Steklov Math. Inst. of RAS, 2024