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I've been working on a problem where I need to know the angle between the tangent vectors of two curves at their intersection point in a flat torus...

Then I thought: Consider two geodesics $\gamma(t)$ and $\beta(t)$ in a flat torus, such that: $\gamma(0)=p=(\varphi _1,\theta_1)$, $\gamma(1)=(\varphi _2,\theta_2)$, $\beta(0)=p$ and $\beta(1)=(\varphi _3,\theta_3)$; wouldn't the angle between their tangent vectors at $p$ the same as the angle between the two "straight lines" that connect those points in this rectangle? If so, I could just get the angle from the usual Euclidean dot product...Is this right?

On a related question: How can I know if in a given riemannian 2-manifold the angles are preserved in the sense I've stated before?

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A smooth map $f:M\to N$ between two Riemannian manifolds $(M,g_M)$ and $(N,g_N)$ is conformal if the pullback metric $f_* g_N$ is of the form $e^u g_M$ where $u$ is some smooth function. This condition expresses the angle-preserving behavior because the scalar multiple $e^u$ cancels out when we calculate angles.

In your case, you are dealing with the quotient map $f:\mathbb R^2\mapsto \mathbb R^2/\mathbb Z^2$ which is a local isometry. Such a map is conformal with $u\equiv 0$. This justifies your computation of angles.

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  • $\begingroup$ Thank you very much. Very concise and clear answer! $\endgroup$ – PML Mar 3 '13 at 17:38

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