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<title>G2-structure</title>

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<span id="openzim-page-title" class="mw-page-title-main"><i>G</i><sub>2</sub>-structure</span>

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<p>

In <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, a <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure</b> is an important type of <a href="G-structure" class="mw-redirect" title="G-structure">G-structure</a> that can be defined on a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>. If <i>M</i> is a smooth manifold of dimension seven, then a G<sub>2</sub>-structure is a reduction of structure group of the <a href="Frame_bundle" title="Frame bundle">frame bundle</a> of <i>M</i> to the <a href="Compact_group" title="Compact group">compact</a>, exceptional <a href="Lie_group" title="Lie group">Lie group</a> <a href="G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>.

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<div class="mw-heading mw-heading2"><h2 id="Equivalent_conditions">Equivalent conditions</h2></div>

<p>The existence of a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> structure on a 7-manifold <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is equivalent to either of the following conditions:

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<ul><li>The first and second <a href="Stiefel%E2%80%93Whitney_class" title="Stiefel–Whitney class">Stiefel–Whitney classes</a> of <i>M</i> vanish.</li>

<li><i>M</i> is <a href="Orientable" class="mw-redirect" title="Orientable">orientable</a> and admits a <a href="Spin_structure" title="Spin structure">spin structure</a>.</li></ul>

<p>It follows that the existence of a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure is much weaker than the existence of a metric of holonomy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>, because a compact 7-manifold of holonomy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> must also have finite fundamental group and non-vanishing first <a href="Pontrjagin_class" class="mw-redirect" title="Pontrjagin class">Pontrjagin class</a>.

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The fact that there might be certain Riemannian 7-manifolds manifolds of holonomy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> was first suggested by <a href="Marcel_Berger" title="Marcel Berger">Marcel Berger</a>'s 1955 classification of possible Riemannian holonomy groups. Although still working in a complete absence of examples, <a href="Edmond_Bonan" title="Edmond Bonan">Edmond Bonan</a> then forged ahead in 1966, and investigated the properties that a manifold of holonomy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> would necessarily have; in particular, he showed that such a manifold would carry a parallel 3-form and a parallel 4-form, and that the manifold would necessarily be Ricci-flat.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, it remained unclear whether such metrics actually existed until <a href="Robert_Bryant_(mathematician)" title="Robert Bryant (mathematician)">Robert Bryant</a> proved a local existence theorem for such metrics in 1984. The first complete (although non-compact) 7-manifolds with holonomy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> were constructed by Bryant and Simon Salamon in 1989.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The first compact 7-manifolds with holonomy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> were constructed by <a href="Dominic_Joyce" title="Dominic Joyce">Dominic Joyce</a> in 1994, and compact <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a <a href="Spin_structure" title="Spin structure">spin structure</a>, and, hence, a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In the same paper, it was shown that certain classes of <a href="G2_manifold" title="G2 manifold"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-manifolds</a> admit a <a href="Contact_geometry#Contact_forms_and_structures" title="Contact geometry">contact structure</a>.

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<div class="mw-heading mw-heading2"><h2 id="Remarks">Remarks</h2></div>

<p>The property of being a <a href="G2_manifold" title="G2 manifold"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-manifold</a> is much stronger than that of admitting a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure. Indeed, being a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-manifold is equivalent to admitting a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure that is <a href="G-structure" class="mw-redirect" title="G-structure">torsion-free</a>.

</p><p>The letter "G" occurring in the phrases "G-structure" and "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-structure" refers to different things. In the first case, G-structures take their name from the fact that arbitrary Lie groups are typically denoted with the letter "G". On the other hand, the letter "G" in "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>" comes from the fact that its Lie algebra is the seventh type ("G" being the seventh letter of the alphabet) in the classification of complex simple Lie algebras by <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>.

</p>

<ul><li><a href="G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="G2_manifold" title="G2 manifold">G<sub>2</sub>-manifold</a>, <a href="Spin(7)_manifold" class="mw-redirect" title="Spin(7) manifold">Spin(7) manifold</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFE._Bonan1966" class="citation cs2">E. Bonan (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", <i>C. R. Acad. Sci. Paris</i>, <b>262</b>: <span class="nowrap">127–</span>129</cite>.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFBryantSalamon1989" class="citation cs2"><a href="Robert_Bryant_(mathematician)" title="Robert Bryant (mathematician)">Bryant, R.L.</a>; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", <i>Duke Mathematical Journal</i>, <b>58</b> (3): <span class="nowrap">829–</span>850, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fs0012-7094-89-05839-0">10.1215/s0012-7094-89-05839-0</a></cite>.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFJoyce2000" class="citation cs2">Joyce, D.D. (2000), <i>Compact Manifolds with Special Holonomy</i>, Oxford Mathematical Monographs, Oxford University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-850601-5</bdi></cite>.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFArikanChoSalur2013" class="citation cs2">Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

</semantics>

</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span>-manifolds", <i>Asian J. Math.</i>, <b>17</b> (2), International Press of Boston: <span class="nowrap">321–</span>334, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1112.2951">1112.2951</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FAJM.2013.v17.n2.a3">10.4310/AJM.2013.v17.n2.a3</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54942812">54942812</a></cite>.</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFBryant1987" class="citation cs2">Bryant, R. L. (1987), "Metrics with exceptional holonomy", <i>Annals of Mathematics</i>, <b>126</b> (2): <span class="nowrap">525–</span>576, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1971360">10.2307/1971360</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1971360">1971360</a></cite>.</li></ul>

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