Ricci tensor equals zero implies flat splace?

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Discussion Overview

The discussion revolves around whether a Ricci tensor equal to zero implies a flat space, exploring the implications in the context of Riemannian and semi-Riemannian manifolds. It includes theoretical considerations and applications in physics.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if a Ricci tensor equal to zero implies flat space, seeking clarification.
  • Another participant argues that a Ricci flat manifold does not necessarily imply a flat Riemannian metric, providing examples of both compact and non-compact Ricci flat manifolds.
  • This participant notes that for compact manifolds, a flat Riemannian metric implies a zero Euler characteristic, referencing the relationship between curvature and the Euler class.
  • A later reply introduces the concept of vacuum in semi-Riemannian geometry, stating that a vacuum can have a Ricci tensor equal to zero while still being curved, using the space above the Earth's surface as an example.
  • Additional resources are provided for further reading on Ricci decomposition and curvature of Riemannian manifolds.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the implications of a Ricci tensor equal to zero and the nature of flat versus curved spaces.

Contextual Notes

The discussion highlights limitations in understanding the relationship between Ricci flatness and the geometry of manifolds, particularly regarding compactness and the implications of curvature.

Abrahamsk8
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Hi, my question is the title, if Ricci tensor equals zero implies flat space? Thanks for your help
 
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If by flat space you mean a flat Riemannian manifold, the answer is no. There are examples of both compact and non-compact Ricci flat manifolds that cannot be given flat Riemannian metrics.

One thing to note is that for compact manifolds,a flat Riemannian metric implies that the Euler characteristic is zero - because the Euler class can be expressed as a polynomial in the curvature 2 form. If you look around you will find Calabi -Yau manifolds with non-zero Euler characteristic.
 
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As a physics application, in the semi-Riemannian case, a vacuum has a Ricci tensor equal to zero. A vacuum doesn't have to be flat. A vacuum be curved by gravity. An example would be the space above the Earth's surface where you're sitting right now.
 

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