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

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SUMMARY

The metric tensor on the manifold ##S^1 \times S^2## can be derived from the standard metrics of the circle and the sphere. The product metric for this manifold is expressed as ##d\theta^2 + d\psi^2 + \sin^2\theta d\phi^2##. This formulation is analogous to the metrics in the plane ##\mathbb{R}^2##, which can be represented as ##dx^2 + dy^2## or in polar coordinates as ##dr^2 + r^2 d\theta^2##. Understanding these relationships is crucial for working with product manifolds in differential geometry.

PREREQUISITES
  • Understanding of differential geometry concepts
  • Familiarity with metric tensors and their properties
  • Knowledge of the standard metrics on ##S^1## and ##S^2##
  • Basic understanding of product manifolds
NEXT STEPS
  • Study the derivation of product metrics in differential geometry
  • Explore the properties of metric tensors on various manifolds
  • Learn about the implications of metric choices on curvature
  • Investigate applications of product metrics in physics and engineering
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Mathematicians, physicists, and students of differential geometry who are interested in understanding the properties and applications of metric tensors on complex manifolds.

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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