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| In [[mathematics]], in particular in [[partial differential equation]]s and [[differential geometry]], an '''elliptic complex''' generalizes the notion of an [[elliptic operator]] to sequences. Elliptic complexes isolate those features common to the [[de Rham cohomology|de Rham complex]] and the [[Dolbeault cohomology|Dolbeault complex]] which are essential for performing [[Hodge theory]]. They also arise in connection with the [[Atiyah-Singer index theorem]] and [[Atiyah-Bott fixed point theorem]].
| | The title of the author is Numbers. Puerto Rico is exactly where he and his spouse live. What I love doing is doing ceramics but I haven't produced a dime with it. Bookkeeping is my profession.<br><br>Feel free to visit my weblog; [http://tti-Ttw.com/index.php?do=/profile-2200/info/ tti-Ttw.com] |
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| ==Definition==
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| If ''E''<sub>0</sub>, ''E''<sub>1</sub>, ..., ''E''<sub>''k''</sub> are [[vector bundles]] on a [[smooth manifold]] ''M'' (usually taken to be compact), then a '''differential complex''' is a sequence
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| :<math>\Gamma(E_0) \stackrel{P_1}{\longrightarrow} \Gamma(E_1) \stackrel{P_2}{\longrightarrow} \ldots \stackrel{P_k}{\longrightarrow} \Gamma(E_k)</math> | |
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| of [[differential operators]] between the [[sheaf (mathematics)|sheaves]] of sections of the ''E''<sub>''i''</sub> such that ''P''<sub>''i''+1</sub> o ''P''<sub>''i''</sub>=0. A differential complex is '''elliptic''' if the sequence of [[symbol of a differential operator|symbols]]
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| :<math>0 \rightarrow \pi^*E_0 \stackrel{\sigma(P_1)}{\longrightarrow} \pi^*E_1 \stackrel{\sigma(P_2)}{\longrightarrow} \ldots \stackrel{\sigma(P_k)}{\longrightarrow} \pi^*E_k \rightarrow 0</math>
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| is [[exact sequence|exact]] outside of the zero section. Here π is the projection of the [[cotangent bundle]] ''T*M'' to ''M'', and π* is the [[pullback bundle|pullback]] of a vector bundle.
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| ==See also== | |
| *[[Chain complex]]
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| [[Category:Differential geometry]]
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| [[Category:Elliptic partial differential equations]]
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| {{differential-geometry-stub}}
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Latest revision as of 13:44, 21 August 2014
The title of the author is Numbers. Puerto Rico is exactly where he and his spouse live. What I love doing is doing ceramics but I haven't produced a dime with it. Bookkeeping is my profession.
Feel free to visit my weblog; tti-Ttw.com