https://diffgeom.subwiki.org/w/index.php?action=history&feed=atom&title=Riemann_curvature_spaceRiemann curvature space - Revision history2026-06-18T17:26:22ZRevision history for this page on the wikiMediaWiki 1.41.2https://diffgeom.subwiki.org/w/index.php?title=Riemann_curvature_space&diff=1214&oldid=prevVipul: 1 revision2008-05-18T19:52:13Z<p>1 revision</p>
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<p><b>New page</b></p><div>==Definition==<br />
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===Given data===<br />
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A real vector space <math>V</math> with an inner product <math>g</math> (making it into the usual Euclidean space).<br />
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===Definition part===<br />
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The Riemann curvature space over <math>V</math>, denoted <math>RC(V)</math>, is the space of multilinear forms on <math>V</math> in 4 variables, satisfying the following identities:<br />
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# <math>R(X,Y,Z,T) = -R(Y,X,Z,T)</math> (viz <math>R</math> is antisymmetric in the first two variables)<br />
# <math>R(X,Y,Z,T) = -R(X,Y,T,Z)</math> (viz <math>R</math> is antisymmetric in the last two variables)<br />
# <math>R(X,Y,Z,T) + R(Y,Z,X,T) + R(Z,X,Y,T) = 0</math> (The first Bianchi identity)<br />
# <math>R(X,Y,Z,T) = R(Z,T,X,Y)</math> (this follows from the above three)<br />
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Note that <math>RC(V)</math> sits naturally as a subspace inside the space <math>Sym^2(Alt^2(V))</math> (follows from the facts (1), (2) and (4).</div>Vipul