Geodesic
Realizability of the $H_k$-distance functions by homology classes of path spaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2010), pp. 18-24

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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. 
@article{IVM_2010_5_a2,
     author = {Yu. V. Ershov and E. I. Yakovlev},
     title = {Realizability of the $H_k$-distance functions by homology classes of path spaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {18--24},
     publisher = {mathdoc},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_5_a2/}
}
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Yu. V. Ershov; E. I. Yakovlev. Realizability of the $H_k$-distance functions by homology classes of path spaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2010), pp. 18-24. http://geodesic.mathdoc.fr/item/IVM_2010_5_a2/