Proving Hodge Decomposition Theorem for Compact Riemannian Manifolds

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In summary, the conversation is about trying to prove the Hodge decomposition theorem for a compact Riemannian manifold. The speaker is looking for a reference or proof for the harmonic part of sigma, which is challenging in infinite dimensional spaces. They mention a heuristic proof involving minimizing the norm w, and suggest checking out the book "Foundations of Differentiable Manifolds and Lie Groups" by Warner for a proof. They also suggest looking at a website for notes on the topic.
  • #1
Haelfix
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Right so I am reviewing some notes I have from a year or two ago, and I am away from my universities library, and there's a few things that are troubling me.

Consider a compact Riemanian manifold M, with a metric. I am looking to prove the Hodge decomposition theorem.

A heuristic proof is that we want to find a unique representative of the cohomology class by minimizing the norm w, where [w] is the class. Basically (im going to update this post the second I figure out how to get latex working) you see that the minimum of the line w + d sigma forms an affine subspace of sigma ^ k and the minimum will be orthogonal to w + d sigma ^ (k-1), that will subsume the coexact part.

The problem I am running into is that this is more or less trivial in finite spaces, but we are dealing with infinite dimensional spaces, so I can't see why the harmonic part of sigma needs to be finite.

Anyone have a good reference to a place where I can find the proof?
 
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  • #2
do you mean a website? one of my favorite references in a book is to warner's foundations of differentiable manifolds and lie groups.
 
  • #3
You could have a look at: http://swc.math.arizona.edu/notes/files/DLSCarlson.pdf [Broken]
 
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  • #4
there ae no proofs at that site. the warner book has a proof.
 

What is Hodge theory?

Hodge theory is a mathematical theory that studies the relationship between the topology and geometry of a smooth manifold. It provides a way to decompose the cohomology groups of a manifold into smaller, more manageable pieces, and has applications in algebraic geometry, differential geometry, and topology.

Why is Hodge theory important?

Hodge theory is important because it allows us to understand the underlying structure and properties of smooth manifolds. It also has many applications in other areas of mathematics, including physics and engineering.

What are the main concepts in Hodge theory?

The main concepts in Hodge theory include differential forms, cohomology groups, harmonic forms, and the Hodge star operator. These concepts are used to define the Hodge decomposition theorem and the Hodge conjecture, which are central results in Hodge theory.

How is Hodge theory used in physics?

In physics, Hodge theory is used to study the behavior of fields on manifolds, such as electromagnetism and general relativity. It provides a way to describe and analyze the interactions between these fields and their underlying geometry.

What are some open problems in Hodge theory?

Some open problems in Hodge theory include the Hodge conjecture, which states that every cohomology class on a smooth projective variety can be represented by a harmonic form, and the Hodge index theorem, which relates the signature of a complex manifold to the intersection form of its complex submanifolds. Additionally, there are many applications of Hodge theory that are still being developed and studied.

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