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Math Help - Help on smooth manifold!

  1. #1
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    Help on smooth manifold!

    Why f: R -> R^2 by f(t) = (t^2, t^3) is not a smooth manifold? Just learned the concept of smooth manifold. Really confused about it.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Help on smooth manifold!

    We cannot find a well defined tangent space at t=0.
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  3. #3
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    Re: Help on smooth manifold!

    It's because the inverse function is not differentiable at the point x = 0.
    A smooth manifold M is a topological space where for every x in M there is a diffeomorphism  f: M \supseteq U \rightarrow V \subseteq \mathbb{R}^n  \ with \ x \in U (which means that for every open subset of M there is a chart (=diffeomorphism) that maps U to an open subset of \mathbb{R}^n ).
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