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    A Pullback of the metric from R3 to S2

    I am looking at this document I do not understand how the author gets 5.12 and 5.13 on page 133. I think the matrix of partials should be the transpose of the one shown. Even so I still can't figure out how you get 5.13. Any help would be appreciated.
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    I Computation of the left invariant vector field for SO(3)

    I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward. I have been looking at these notes: https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...
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    A Tangent and Cotangent Bases

    Thank you. Now I understand.
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    A Tangent and Cotangent Bases

    I am trying to figure how one arrives at the following: dxμ∂ν = ∂xμ/∂xν = δμν Where, dxμ is the gradient of the coordinate functions = basis of cotangent space ∂ν = basis of tangent space I know that dual vectors 'eat' vectors to produce scalars. Is this demonstrated by absorbing d into ∂...
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    A Confusion regarding conformal transformations

    Thanks. I think I am almost there. By convention Ω = exp(2f). Both the CT and the WT are applicable to d ≥ 2 but for d = 2, f = f(z) and is holomorphic. The scale factor becomes |df/dz|2. Correct?
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    A Confusion regarding conformal transformations

    OK, I have modified my original questions. I think (hope) I now have a better understanding. Perhaps somebody could critique this. However, I am still a little confused about the relationship between λ and Ω and the difference in exponents. Thanks. Conformal Manifolds: A manifold, M, is...
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    A Confusion regarding conformal transformations

    I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far. 1. Under a conformal transformation the metric changes by: g' -> Ω2g 2. Under a Weyl transformation the metric changes by: g' -> exp(-2f)g 3. Any 2D Riemann manifold is locally conformally...
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    Product of dyadic and a vector

    Flat space. I am trying to show that the gradient of a contravariant vector is a covariant vector. I understand how to show this for a scalar, but not sure how to extend this to vectors/tensors.
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    Product of dyadic and a vector

    I have: dVμ = (∂Vμ/∂xη)dxη where Vμ is a contravariant vector field I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dxη a scalar and how do I write this is compact form. I know how this works for scalars but am not clear when tensors are involved.
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