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Math Help - Does a diffeomorphism between manifolds induce a diffeomorphism on cotangent space?

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    Does a diffeomorphism between manifolds induce a diffeomorphism on cotangent space?

    I'm reading a book, and this seems to be true, but they don't say why, and I'm not sure.
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    MHF Contributor Drexel28's Avatar
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    Re: Does a diffeomorphism between manifolds induce a diffeomorphism on cotangent spac

    I'm a little confused. While it is true that the cotangent space of a manifold naturally carries the structure of a smooth manifold (it's just a finite dimensional vector space) this is a strange question. I believe what you meant to ask is whether or not a diffeomorphism induces a linear isomorphism on the cotangent spaces. This is in fact true. In particular, let's suppose that we have a diffeomorphism f:M\to N. I think you know then that the derivative df induces a linear isomorphism T_xM\to T_{f(x)}N for each point x\in M. Dualizing this, you get an isomorphism df^\ast(x):T_x^\ast N\to T_x^\ast M, which is the desired isomorphism.
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    Re: Does a diffeomorphism between manifolds induce a diffeomorphism on cotangent spac

    small addendum: linear maps are always differentiable, and if a linear map is invertible, it's inverse is also a linear map, and thus differentiable, so a linear isomorphism is a diffeomorphism (not a very interesting one, though).
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    MHF Contributor Drexel28's Avatar
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    Re: Does a diffeomorphism between manifolds induce a diffeomorphism on cotangent spac

    I'd like to add, that if your manifold has a Riemannian structure there is a covariant way to do this. Namely, take the isomorphism df(x):T_xM\to T_{f(x)}N and consider the identification T_xM\cong T_X^\ast M given by your metric.
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