## 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

Zbl 1118.53028
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