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I'm taking a course on general relativity and I did a problem that I have now found out I have no idea how to interpret.

The surface of a paraboloid has the metric

$$ds^2=(1+r^2)dr^2+r^2d\theta^2$$

I was asked to do parallel transport of the vector $\mathbf{V}=V\hat{e_r}$ along a circle of radius $r_o$.

I know how to do the mechanics of this problem but I have these questions:

What would've happened if I wanted to transport a vector with a component along the $\mathbf{z}$ direction? The calculation of the Riemann tensor, christoffel symbols etc, assume I live in a 2D place and my indices only have 2 possible values. What would happen here?

Why exactly is it that $e_r$ is the same vector as the one in polar coordinates for Euclidean space? How does my space inherit the base vectors of a completely different space?

Perhaps my questions are really basic, but I have seen differential geometry in a very algorithmic fashion without any deep interpretation.

Thanks.

DLV
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  • Notice the "hat" on your vector. Here you take a vector in the plane, but using local coordinates you look at its image, its corresponding tangent vector, which will in fact have vertical component. The reason the plane seems to inherit the base vectors of a different (curved) space is that you forget about the plane's Euclidean inner product. $i$ and $j$ do not map to orthogonal images on the manifold. Thus, only topologically a manifold resembles a Euclidean open set, but not geometrically -- that is when we talk about metric on our manifold. Hope this helps. – Behnam Esmayli May 13 '16 at 04:02

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