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Examples of indefinite globally framed \(f\)-structures on compact Lie groups. (English) Zbl 1265.53037

Let \((M,g)\) be a \((2n+r)\)-dimensional semi-Riemannian manifold equipped with a \((1,1)\)-tensor \(\varphi\), \(r\) vector field \(\xi_i\), and \(r\) \(1\)-forms \(\eta^i\) such that \(\eta^i(\xi_j)=\varepsilon_i\delta^i_j\), \(\varphi^2=-\varepsilon_j\text{id}+\sum\limits_{i=1}^r\eta^i\otimes\xi_i\), \(g(\varphi X,\varphi Y)=g(X,Y)-\sum\limits_{i=1}^r\varepsilon_i\eta^i(X)\eta^i(Y)\), and \(\eta^i(X)=\varepsilon_ig(X,\xi_i)\), where \(\varepsilon_i=\pm1\) according to whether \(\xi_i\) is spacelike or timelike. Then \((M,g,\varphi,\xi_i,\eta^i)\) is called an indefinite \(f\)-manifold with parallelizable kernel. If the \(2\)-form \(F\) defined by \(F(X,Y)=g(\varphi X,Y)\) is closed and \(N(X,Y)=N_\varphi(X,Y)+d\eta^i(X,Y)\xi_i=0\), where \(N_\varphi\) is the Nijenhuis tensor, then \(M\) is an indefinite \({\mathcal K}\)-manifold, and if \(d\eta^i=F\), then \(M\) is an indefinite \({\mathcal S}\)-manifold, and if all \(\eta^i\) are closed, then \(M\) is an indefinite \({\mathcal C}\)-manifold. Let \({\mathcal F}\) be the distribution spanned by \(\xi_i\)’s. In [Trans. Am. Math. Soc. 181, 175–184, (1973; Zbl 0276.53026)], D. E. Blair et al. considered metric \(f\)-manifolds with parallelizable kernel and they showed that for a connected and compact metric \(f\)-manifold \((M,g,\varphi,\xi_i,\eta^i)\) with parallelizable kernel, the manifold \(M\) is the total space of a principal toroidal bundle over a complex manifold \(N^{2n}=M/{\mathcal F}\), and if \(M\) is a \({\mathcal K}\)-manifold, then \(N^{2n}\) is a Kähler manifold.
In this paper, the authors extend the above result to the indefinite case. They prove that if \((M,g,\varphi,\xi_i,\eta^i)\) is a connected and compact indefinite \(f\)-manifold with parallelizable kernel such that all \(\xi_i\)’s are Killing, then \(M\) is the total space of a principal toroidal bundle over a Hermitian or indefinite Hermitian manifold \(N^{2n}=M/{\mathcal F}\), and if \(M\) is an indefinite \({\mathcal K}\)-manifold, then \(N^{2n}\) is either a Kähler manifold or an indefinite Kähler manifold. In this way they obtain examples of compact indefinite \({\mathcal S}\)-manifolds. Then, the authors define an indefinite \({\mathcal S}\)-structure on the Lie group \(\mathrm{U}(2)\) with a Lorentz left-invariant metric and construct commutative diagrams involving semi-Riemannian submersions and Hopf fibrations to prove that \(\mathrm{U}(2)\) with such a structure is foliated by Reinhart lightlike hypersurfaces.
Finally, they consider a normal indefinite \(f\)-structure with parallelizable kernel on the Lie group \(\mathrm{U}(4)\) and prove that it is not an indefinite \({\mathcal S}\)-structure.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53Z05 Applications of differential geometry to physics

Citations:

Zbl 0276.53026
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