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As is known, the vector space $V\oplus V^\ast$ admits the natural symmetric and skew-symmetric bilinear forms $$\langle X+\xi,Y+\eta\rangle|_\pm:=\frac 1 2 (\xi(Y) \pm \eta(X)).$$

I am interested in collecting results concerning these bilinear forms and their applications. They were used for example in

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Both these forms are indefinite (trace zero), so perhaps they should not be called inner products in the title? – Noah Stein Feb 28 '13 at 15:25
This is a local version of a global result, namely that cotangent bundles are symplectic manifolds, and this gets used in mathematical physics. – Qiaochu Yuan Feb 28 '13 at 18:30

Search for Dirac structures or Courant algebroids in MathSciNet: These are common generalizations of symplectic and Poisson structures and use the symmetric bilinear form on $TM\times_M T^*M$ on a manifold: Namely, the graph of a symplectic structure as well as the graph of a Poisson structure are maximal isotropic subbundles, with further properties.

There is a lot of literature on them now.

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Thank you. Indeed, Dirac structures are also mentioned in the works on Generalized complex geometry. – Cristi Stoica Mar 1 '13 at 8:22

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