Paracompactness and metrizable manifolds

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SUMMARY

A manifold is metrizable with a Riemannian metric if and only if it is paracompact, as established in the discussion. The proof can be found in "Foundations of Differential Geometry" by Kobayashi and Nomizu, Volume I, although it is noted to be complex. An alternative, more accessible proof is available in "Topology" by James Munkres, which, while easier, still spans several pages. Both resources are essential for understanding this theorem in differential geometry.

PREREQUISITES
  • Understanding of Riemannian metrics
  • Familiarity with paracompactness in topology
  • Knowledge of differential geometry principles
  • Basic concepts from topology, particularly from Munkres' "Topology"
NEXT STEPS
  • Read "Foundations of Differential Geometry" by Kobayashi and Nomizu for a detailed proof
  • Study "Topology" by James Munkres for an alternative proof of the theorem
  • Explore the implications of paracompactness in manifold theory
  • Investigate other theorems related to metrization in differential geometry
USEFUL FOR

Mathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to deepen their understanding of the relationship between paracompactness and metrizability in manifolds.

aleazk
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Hi, where I can find a proof of the theorem that establishes that a manifold is metrizable (with a riemannian metric) if and only if is paracompact?.
 
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Wow, I found a proof in Kobayashi and Nomizu, Vol I, and it seems pretty hard :frown:. I think that simply I will have to accept the result :cry:
 
Hi aleazk! :smile:

Try "topology" of Munkres, it gives an easy proof of the result (but the proof is still several pages long :smile: )
 

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