Manifolds with vectorial torsion

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• 作者：Ilka Agricola^a ; ¹ ; ^{agricola@mathematik.uni-marburg.de" class="auth_mail" title="E-mail the corresponding author} ; Margarita Kraus^b ; ^{mkraus@mathematik.uni-mainz.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 53 C 25 ; secondary ; 81 T 30
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：45
• 期：Complete
• 页码：130-147
• 全文大小：443 K

文摘

The present note deals with the properties of metric connections ∇ with vectorial torsion V   on semi-Riemannian manifolds (Mⁿ,g)den">de">$(M^n,g)$. We show that the ∇-curvature is symmetric if and only if de365ce0cf5bda06fc5f14a6085" title="Click to view the MathML source">V^♭den">de">$V^{♭}$ is closed, and that V^⊥den">de">$V^{\perp }$ then defines an (n−1)den">de">$(n-1)$-dimensional integrable distribution on de7ef1ee7f41b74fc320c0769588" title="Click to view the MathML source">Mⁿden">de">$M^n$. If the vector field V is exact, we show that the V-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature—for example, a V-Ricci-flat connection with vectorial torsion in dimension 4, explaining some constructions occurring in general relativity. Finally, we investigate the Dirac operator D of a connection with vectorial torsion. We prove that for exact vector fields, the V-Dirac spectrum coincides with the spectrum of the Riemannian Dirac operator. We investigate in detail the existence of V-parallel spinor fields; several examples are constructed. It is known that the existence of a V  -parallel spinor field implies dV^♭=0den">de">$d{V}^{♭}=0$ for n=3den">de">$n=3$ or n≥5den">de">$n\ge 5$; for n=4den">de">$n=4$, this is only true on compact manifolds. We prove an identity relating the V-Ricci curvature to the curvature in the spinor bundle. This result allows us to prove that if there exists a nontrivial V  -parallel spinor, then Ric^V=0den">de">${\mathrm{Ric}}^V=0$ for n≠4den">de">$n\ne 4$ and Ric^V(X)=XdV^♭den">de">${\mathrm{Ric}}^V(X)=X\text{<inline-figure baseline="0.0"><link locator="fx001" type="simple" href="pii:S0926-2245(16)00013-9/fx001"></inline-figure>}d{V}^{♭}$ for n=4den">de">$n=4$. We conclude that the manifold is conformally equivalent either to a manifold with Riemannian parallel spinor or to a manifold whose universal cover is the product of Rden">de">$\mathbb{R}$ and an Einstein space of positive scalar curvature. We also prove that if dV^♭=0den">de">$d{V}^{♭}=0$, there are no non-trivial ∇-Killing spinor fields.