I have a conceptual question associated with one of the worked examples in my notes. The question is:(adsbygoogle = window.adsbygoogle || []).push({});

'Let ##\nabla## and ##\nabla^*## be connections on a manifold ##M##. Show that ##H(X,Y) = \nabla_X Y - \nabla_X^* Y## where ##X,Y## are vector fields defines a (1,2) tensor on M.

To show it is a tensor, one needs to check linearity of the arguments X and Y. This is clear, but I don't understand why these arguments show that H is necessarily of rank (1,2)? The notation H(X,Y) seems to suggest it is a (1,1) tensor since the map H acts on one covector and one vector argument. The solution says we may define a (1,2) tensor by ##(\lambda, X,Y) \mapsto \langle \lambda, H(X,Y)\rangle##, but I am not sure why it must be (1,2) and why this means H is of rank (1,2)?

Thanks!

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# Why $H$ is a (1,2) tensor field?

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