×

zbMATH — the first resource for mathematics

Geometric realizations of curvature models by manifolds with constant scalar curvature. (English) Zbl 1191.53017
Let \(V\) be a real vector space of dimension \(n\) and let \(A\) be an algebraic curvature tensor on \(V\). We recall that \({\mathcal M} = (V,\langle\;,\;\rangle, A)\) is a curvature model if \(A\) is an algebraic curvature tensor on \(V\) and if \(\langle\;,\;\rangle\) is a non-degenerate bilinear form of signature \((p,q)\) on \(V\).
It is well known that given a curvature model \(\mathcal M\), there exist a real analytic pseudo-Riemannian manifold \(M\) and a point \(p\) of \(M\) such that \(M\) has a curvature model isomorphic to \(\mathcal M\). The authors extend this result to the class of manifolds with constant scalar curvature showing that any pseudo-Riemannian curvature model can be geometrically realized by a pseudo-Riemannian manifold with constant scalar curvature. Moreover, they prove that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and constant \(*\)-scalar curvature.
Reviewer: Anna Fino (Torino)

MSC:
53B20 Local Riemannian geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aubin, T., Équations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire, J. math. pures appl., 55, 269-296, (1976) · Zbl 0336.53033
[2] Blažić, N., Natural curvature operators of bounded spectrum, Differential geom. appl., 24, 563-566, (2006) · Zbl 1106.53014
[3] Martín Cabrera, F.; Swann, A., Almost Hermitian structures and quaternionic geometries, Differential geom. appl., 21, 199-214, (2004) · Zbl 1062.53034
[4] del Rio, H.; Simanca, The Yamabe problem for almost Hermitian manifolds, J. geom. anal., 13, 185-203, (2003) · Zbl 1048.53019
[5] Díaz-Ramos, J.C.; García-Río, E.; Vázquez-Lorenzo, R., New examples of Osserman metrics with nondiagonalizable Jacobi operators, Differential geom. appl., 24, 433-442, (2006) · Zbl 1099.53047
[6] Cortés-Ayaso, A.; Díaz-Ramos, J.C.; García-Río, E., Four-dimensional manifolds with non-degenerated self-dual Weyl curvature tensor, Ann. global anal. geom., 34, 185-193, (2008) · Zbl 1181.53030
[7] Cruceanu, V.; Fortuny, P.; Gadea, P.M., A survey on paracomplex geometry, rocky mountain, J. math., 26, 83-115, (1996) · Zbl 0856.53049
[8] Díaz-Ramos, J.C.; García-Río, E.; Vázquez-Lorenzo, R., Osserman metrics on Walker 4-manifolds equipped with a para-Hermitian structure, Mat. contemp., 30, 91-108, (2006) · Zbl 1152.53311
[9] Evans, L.C., Partial differential equations, Graduate texts in mathematics, ISBN: 0-8218-0772-2, vol. 19, (1998), American Mathematical Society Providence, RI, xviii+662 pp
[10] Falcitelli, M.; Farinola, A.; Salamon, S., Almost-Hermitian geometry, Differential geom. appl., 4, 259-282, (1994) · Zbl 0813.53044
[11] Gadea, P.M.; Oubiña, J.A., Homogeneous pseudo-riemannian structures and homogeneous almost para-Hermitian structures, Houston J. math., 18, 449-465, (1992) · Zbl 0760.53029
[12] García-Río, E.; Kupeli, D.; Vázquez-Abal, M.E.; Vázquez-Lorenzo, R., Osserman affine connections and their Riemannian extensions, Differential geom. appl., 11, 145-153, (1999) · Zbl 0940.53017
[13] Gray, A., Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku math. J., 28, 601-612, (1976) · Zbl 0351.53040
[14] Ivanov, S.; Zamkovoy, S., Parahermitian and paraquaternionic manifolds, Differential geom. appl., 23, 205-234, (2005) · Zbl 1115.53022
[15] Kamada, H., Neutral hyperkähler structures on primary Kodaira surfaces, Tsukuba J. math., 23, 321-332, (1999) · Zbl 0948.53023
[16] Nikolayevsky, Y., Two theorems on Osserman manifolds, Differential geom. appl., 18, 239-253, (2003) · Zbl 1051.53039
[17] Schoen, R., Conformal deformation of a Riemannian metric to constant scalar curvature, J. differential geom., 20, 479-495, (1984) · Zbl 0576.53028
[18] Trudinger, N., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. scuola norm. sup. Pisa, 22, 265-274, (1968) · Zbl 0159.23801
[19] Yamabe, H., On a deformation of Riemannian structures on compact manifolds, Osaka math. J., 12, 21-37, (1960) · Zbl 0096.37201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.