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Geometric structures of vectorial type. (English) Zbl 1106.53033

Let \((M^n,g)\) be an oriented Riemannian manifold, \(\mathcal{F}(M^n)\) its Riemannian frame bundle, which is a principal \(SO(n)\)-bundle, \(G\) a fixed subgroup of \(SO(n)\) and \(\mathcal{R}\subset\mathcal{F}(M^n)\) a \(G\)-structure. Let \(Z\) be the Levi-Civita connection on \(\mathcal{F}(M^n)\). \(Z\) is a \(1\)-form on \(\mathcal{F}(M^n)\) with values in the Lie algebra \(so(n)=\Lambda^2(\mathbb{R}^n)\). Decompose \(so(n)=\mathfrak{g}\oplus \mathfrak{m}\) into the Lie algebra \(\mathfrak{g}\) of \(G\) and its orthogonal complement \(\mathfrak{m}\). With respect to this decomposition, \(Z| _{T(\mathcal{R})}:=Z^*\oplus \Gamma\), where \(Z^*\) is a connection in the principal \(G\)-bundle \(\mathcal{R}\) and \(\Gamma\) is a \(1\)-form on \(M^n\) with values in the associated bundle \(\mathcal{R}\times_G\mathfrak{m}\) called intrinsic torsion of \(\mathcal{R}\). The different types of \(G\)-structures on a Riemannian manifold correspond to the irreducible \(G\)-components of the representation \(\mathbb{R}^n\otimes\mathfrak{m}\).
The mapping \(\Theta:\mathbb{R}^n\to \mathbb{R}^n\times\mathfrak{m}\), \(\Theta(u):=\sum^n_{i=1}e_i\otimes pr_{\mathfrak{m}}(e_i\wedge u)\), shows that the space \(\mathbb{R}^n\otimes\mathfrak{m}\) contains \(\mathbb{R}^n\) in a natural way. \(\mathcal{R}\) is called of vectorial type (or \(\mathcal{W}_4\)-structure) if the element \(\Gamma\in\mathbb{R}^n\otimes\mathfrak{m}\) belongs to \(\mathbb{R}^n\). If the structure group \(G\) preserves a non-degenerate differential \(k\)-form or if \(G\) lifts into the spin group \(\text{Spin}(n)\) and admits a \(G\)-invariant algebraic spinor \(\Psi\in\Delta_n\) in the \(n\)-dimensional spin representation \(\Delta_n(n\geq 5)\), the authors show that for any \(G\)-structure of vectorial type, \(\Gamma\) is a closed \(1\)-form. From the point of view of Weyl geometry, this last result means that any \(G\)-structure with a fixed spinor on a compact manifold \(M^n\) induces a Weyl-Einstein geometry with a closed form \(\Gamma\) \((n\geq 5)\).
The authors prove that – up to a conformal change of the metric – compact \(G\)-structures of vectorial type admitting a parallel spinor are locally conformal to products of \(\mathbb{R}\) with Einstein manifolds with positive scalar curvature admitting a real Riemannian Killing spinor. This splitting result generalizes and unifies a result established by S. Ivanov, M. Parton, P. Piccinni [Math. Res. Lett. 13, No. 2–3, 167–177 (2006; Zbl 1118.53028)] in dimensions \(n=7,8\) only. \(G\)-structures of vectorial type admitting a characteristic connection \(\nabla^c\) (that is a \(G\)-connection with totally skew-symmetric torsion tensor) are also investigated. If \(\mathcal{R}\) is such a \(G\)-structure and \(\nabla^c\Gamma=0\) holds, \(\mathcal{R}\) is called a generalized Hopf structure. In this case it is shown that \(\Gamma\) is a Killing vector field and for the structure groups \(G=G_2\), Spin(7) and \(U(n)\), \(n\geq 3\), it is even parallel.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory

Citations:

Zbl 1118.53028
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References:

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