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

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Discussion Overview

The discussion revolves around finding the metric tensor on the product manifold ##S^1 \times S^2##, exploring the concept of metrics on manifolds, and examining the relationship between metrics on different spaces.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to find the metric tensor on ##S^1 \times S^2##.
  • Another participant asserts that there is no unique metric tensor on a manifold, stating that infinitely many metrics exist and suggesting the use of the product metric derived from the standard metrics on the circle and sphere.
  • Several participants inquire about the method for taking the product metric of the metrics on the circle and sphere.
  • Participants mention the metric in the plane ##\mathbb{R}^2##, providing two forms: ##dx^2 + dy^2## and ##dr^2 + r^2 d\theta^2##.
  • One participant elaborates that the plane ##\mathbb{R}^2## can be viewed as the product ##\mathbb{R} \times \mathbb{R##, with the metrics on each factor being ##dx^2## and ##dy^2##.
  • A later reply proposes that for ##S^1 \times S^2##, the metric could be expressed as ##d\theta^2 + d\psi^2 + \sin^2\theta d\phi^2##.

Areas of Agreement / Disagreement

Participants express differing views on the existence and uniqueness of the metric tensor, with some asserting the variability of metrics on manifolds while others seek clarification on specific metrics for ##S^1 \times S^2##. The discussion remains unresolved regarding the exact formulation of the metric tensor.

Contextual Notes

The discussion includes assumptions about the standard metrics on the circle and sphere, as well as the nature of product metrics. There are unresolved details regarding the specific steps to derive the product metric for ##S^1 \times S^2##.

Onyx
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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##?
 
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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.
 
How do I take the product metric of the circle and sphere metrics?
 
Onyx said:
How do I take the product metric of the circle and sphere metrics?
What is the metric in the plane ##\mathbb R^2##?
 
##dx^2+dy^2## or ##dr^2+r^2d\theta^2##.
 
Onyx said:
##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.
 
martinbn said:
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##.
 

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