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Ellipsoids are examples of [[algebraic varieties]]; and so, for general rank, symmetric tensors, in the guise of [[homogeneous polynomial]]s, are used to define [[projective varieties]], and are often studied as such.
Given a [[Riemannian manifold]] <math>(M,g)</math> equipped with its Levi-Civita connection <math>\nabla</math>, the [[Riemann curvature tensor#Coordinate expression|covariant curvature tensor]] is a symmetric order 2 tensor over the vector space <math display="inline">V = \Omega^2(M) = \bigwedge^2 T^*M</math> of differential 2-forms. This corresponds to the fact that, viewing <math>R_{ijk\ell} \in (T^*M)^{\otimes 4}</math>, we have the symmetry <math>R_{ij\, k\ell} = R_{k\ell\, ij}</math> between the first and second pairs of arguments in addition to antisymmetry within each pair: <math>R_{jik\ell} = - R_{ijk\ell} = R_{ij\ell k}</math>.<ref>{{Cite book |last=Carmo |first=Manfredo Perdigão do |url=https://www.worldcat.org/oclc/24667701 |title=Riemannian geometry |date=1992 |publisher=Birkhäuser |others=Francis J. Flaherty |isbn=0-8176-3490-8 |___location=Boston |oclc=24667701}}</ref>
==Symmetric part of a tensor==
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