Reduced $L_{q,p}$-Cohomology of Some Twisted Products

Golʼdshtein, Vladimir; Kopylov, Yaroslav

Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 151-169.

On établit des résultats d’annulation de la cohomologie $L_{q,p}$ réduite pour les produits twistés, une généralisation des produits tordus dans le cas de $q\le p$. Le résultats obtenus sont des généralisations de certains résultats par Gol${}^{\text{'}}$dshtein, Kuz${}^{\text{'}}$minov et Shvedov sur la cohomologie $L_p$ des cylindres tordus. Une des observations principales est la trivialité de la cohomolgie en dimension “moyenne” pour une large classe de variétés.

Vanishing results for reduced $L_{q,p}$-cohomology are established in the case of twisted products, which are a generalization of warped products. Only the case $q\le p$ is considered. This is an extension of some results by Gol${}^{\text{'}}$dshtein, Kuz${}^{\text{'}}$minov and Shvedov about the $L_p$-cohomology of warped cylinders. One of the main observations is the vanishing of the “middle-dimensional” cohomology for a large class of manifolds.

Classification : 58A10, 58A12
Keywords: differential form, reduced $L_{q,p}$-cohomology, twisted cylinder
Mot clés : forme différentielle, cohomologie $L_{q,p}$ réduite, cylindre twisté

[1] Bishop, Richard L.; O’Neill, Barrett Manifolds of negative curvature, Trans. Am. Math. Soc., Volume 145 (1969), pp. 1-49 | DOI

[2] Boulal, A.; Djaa, Nour El Houda; Djaa, Mustapha; Ouakkas, Seddik Harmonic maps on generalized warped product manifolds, Bull. Math. Anal. Appl., Volume 4 (2012) no. 1, pp. 156-165

[3] Boulal, A.; Djaa, Nour El Houda; Zagane, A. Generalized warped product manifolds and biharmonic maps, Acta Math. Univ. Comen. New Ser., Volume 81 (2012) no. 2, pp. 283-298

[4] Chen, Bang-yen Geometry of Submanifolds and Its Applications, Science University of Tokyo, Tokyo, 1981, iii+96 pages

[5] Falcitelli, Maria A class of almost contact metric manifolds and double-twisted products, Math. Sci. Appl. E-Notes, Volume 1 (2013) no. 1, pp. 36-57

[6] Fernández-López, Manuel; García-Río, Eduardo; Kupeli, Demir N.; Ünal, Bülent A curvature condition for a twisted product to be a warped product, Manuscr. Math., Volume 106 (2001) no. 2, pp. 213-217 | DOI

[7] Golʼdshtejn, Vladimir M.; Kuzʼminov, Vladimir I.; Shvedov, Igor A. Reduced $L_p$-cohomology of warped cylinders, Sib. Math. J., Volume 31 (1990) no. 5, pp. 716-727 | DOI

[8] Golʼdshtejn, Vladimir M.; Troyanov, Marc Sobolev inequalities for differential forms and $L_{q,p}$-cohomology, J. Geom. Anal., Volume 16 (2006) no. 4, pp. 597-632 | DOI

[9] Golʼdshtejn, Vladimir M.; Troyanov, Marc The Hölder–Poincaré duality for $L_{q,p}$-cohomology, Ann. Global Anal. Geom., Volume 41 (2012) no. 1, pp. 25-45 | DOI

[10] Kim, Byung Hak; Jung, Seoung Dal; Kang, Tae Ho; Pak, Hong Kyung Conformal transformations in a twisted product space, Bull. Korean Math. Soc., Volume 42 (2005) no. 1, pp. 5-15 | DOI

[11] Kopylov, Yaroslav $L_{p,q}$-cohomology and normal solvability, Arch. Math., Volume 89 (2007) no. 1, pp. 87-96 | DOI

[12] Kuzʼminov, Vladimir I.; Shvedov, Igor A. On normal solvability of the exterior differentiation on a warped cylinder, Sib. Math. J., Volume 34 (1993) no. 1, pp. 73-82 | DOI

[13] Kuzʼminov, Vladimir I.; Shvedov, Igor A. On normal solvability of the operator of exterior derivation on warped products, Sib. Math. J., Volume 37 (1996) no. 2, pp. 276-287 | DOI

[14] Ponge, Ralf; Reckziegel, Helmut Twisted products in pseudo-Riemannian geometry, Geom. Dedicata, Volume 48 (1993) no. 1, pp. 15-25 | DOI