Manifolds with vectorial torsion

详细信息    查看全文

• 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si1.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=cccd7b71975a5a528bae3afd26af1e5c" title="Click to view the MathML source">(Mⁿ,g)class="mathContainer hidden">class="mathCode">$(M^n,g)$. We show that the ∇-curvature is symmetric if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si2.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=69f4fde365ce0cf5bda06fc5f14a6085" title="Click to view the MathML source">V^♭class="mathContainer hidden">class="mathCode">$V^{♭}$ is closed, and that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si116.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=012b33a793e469d49855b5fbec6a6709" title="Click to view the MathML source">V^⊥class="mathContainer hidden">class="mathCode">$V^{\perp }$ then defines an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si107.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=d265277c79fcaaf42b04497f58b23418" title="Click to view the MathML source">(n−1)class="mathContainer hidden">class="mathCode">$(n-1)$-dimensional integrable distribution on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si5.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=32d5de7ef1ee7f41b74fc320c0769588" title="Click to view the MathML source">Mⁿclass="mathContainer hidden">class="mathCode">$M^n$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si317.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=a052c8faa458d4936fee29727496a8e6" title="Click to view the MathML source">dV^♭=0class="mathContainer hidden">class="mathCode">$d{V}^{♭}=0$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si7.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=07b53f1090f15acd9e523d359a9c61b0" title="Click to view the MathML source">n=3class="mathContainer hidden">class="mathCode">$n=3$ or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si8.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=183844b07ed73943ac8d87eb1ff140c1" title="Click to view the MathML source">n≥5class="mathContainer hidden">class="mathCode">$n\ge 5$; for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si268.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=57f3074753daafd0208d85f580e6f4fe" title="Click to view the MathML source">n=4class="mathContainer hidden">class="mathCode">$n=4$, 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si10.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=0ea6b283e0cc0885498f00c8e94ad72c" title="Click to view the MathML source">Ric^V=0class="mathContainer hidden">class="mathCode">${\mathrm{Ric}}^V=0$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si11.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=67c2ba768af970192d198756d5a90a20" title="Click to view the MathML source">n≠4class="mathContainer hidden">class="mathCode">$n\ne 4$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si12.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=3fe542bf34c0f42dfdc02e048e641ca3" title="Click to view the MathML source">Ric^V(X)=Xclass="imgLazyJSB inlineImage" height="10" width="8" alt="Full-size image (<1 K)" title="Full-size image (<1 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0926224516000139-fx001.gif">dV^♭class="mathContainer hidden">class="mathCode">${\mathrm{Ric}}^V(X)=X\text{<inline-figure baseline="0.0"><link locator="fx001" type="simple" href="pii:S0926-2245(16)00013-9/fx001"></inline-figure>}d{V}^{♭}$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si268.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=57f3074753daafd0208d85f580e6f4fe" title="Click to view the MathML source">n=4class="mathContainer hidden">class="mathCode">$n=4$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si13.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=b2419e119886586411b2615ab6173b16" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathbb{R}$ and an Einstein space of positive scalar curvature. We also prove that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224516000139&_mathId=si317.gif&_user=111111111&_pii=S0926224516000139&_rdoc=1&_issn=09262245&md5=a052c8faa458d4936fee29727496a8e6" title="Click to view the MathML source">dV^♭=0class="mathContainer hidden">class="mathCode">$d{V}^{♭}=0$, there are no non-trivial ∇-Killing spinor fields.