Metric Tensor on ##S^1## x ##S^2##

[Onyx]

TL;DR

How do I find the metric tensor on $S^1$ x $S^2$?

How do I find the metric tensor on $S^1$ x $S^2$?

 

[martinbn]

There is no such thing as the metric tensor. On a given manifold there are infinitely many metrics. For example if you take the standard metrics on the circle and on the sphere you can take the product metric on your manifold.

 

[Onyx]

How do I take the product metric of the circle and sphere metrics?

 

[martinbn]

How do I take the product metric of the circle and sphere metrics?

What is the metric in the plane $\mathbb R^2$?

 

[Onyx]

$dx^2+dy^2$ or $dr^2+r^2d\theta^2$.

 

[martinbn]

$dx^2+dy^2$ or $dr^2+r^2d\theta^2$.

Well, the plane $\mathbb R^2$ is the product $\mathbb R \times \mathbb R$ and the $dx^2$ and $dy^2$ are the metrics on each factor.

 

[Onyx]

Well, the plane $\mathbb R^2$ is the product $\mathbb R \times \mathbb R$ and the $dx^2$ and $dy^2$ are the metrics on each factor.

Well then I suppose for $S^1 x S^2$ it would be $d\theta^2+d\psi^2+sin^2\theta d\phi^2$.