Let $\omega$ a 1-form on a riemannian manifold $(M,g)$, and for a point $x\in M$, there is a notation: $|\omega_x|_g$, what does $|\omega_x|_g$ mean?
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It's the norm of the linear functional $\omega_x$ with respect to the scalar product induced on the cotangent bundle by the Riemannian metric. E.g. in coordinates, $$|\omega_x|^2= \sum_{i,j} g^{ij}(x)\omega_i(x) \omega_j(x)$$ where, in commonly used notation, $(g^{ij})$ is the inverse of $(g_{ij})$
Thomas
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What is $\omega_i(x)$? – Antoine Mar 18 '14 at 16:11
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@Antoine The $i$-th component of the differential form $\omega$ in the coordinate system of choice, at $x$. – Thomas Mar 18 '14 at 17:12
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For example, in the local coordinate, $\omega_x=\sum \omega_i(x)dx_i$. Is it like this? – Antoine Mar 18 '14 at 18:31
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Yes, exactly. The convention is, however, to write $dx^i$ instead of $dx_i$, so that the summation convention applies. – Thomas Mar 18 '14 at 19:08
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Thanks. I have an another related question: let $\omega_x=d_xf$, $f:M\to M$, how to write $|d_xf|$? – Antoine Mar 18 '14 at 19:50
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Just use $\omega_i(x) = \frac{\partial f}{\partial x^i}$(x) – Thomas Mar 18 '14 at 20:02