Hello! I am a bit confused about how the metric transforms vector into one forms. If we have a 2-sphere and we take a point on its surface, we have a tangent plane there on which we define vectors at that point. A one form at that point is associated to a vector at that point through the metric on the sphere i.e. ##\omega_\mu = g_{\mu \nu} A^\nu##. However, if I understood this correctly, the tangent space is ##R^2##, in which the metric is ##diag(1,1)##. So if both the vectors and the one-forms (or tensors in general) are defined at a point, so in the tangent space at that point, why are they different than the ones in ##R^2## i.e. if in ##R^2## we would use the ##diag(1,1)## metric to go from vectors to one forms, why in the tangent space of a point on a sphere, which is also ##R^2##, we use the metric of the sphere and not ##diag(1,1)##? Thank you!(adsbygoogle = window.adsbygoogle || []).push({});

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# I Metric and one form

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