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Michał Oszmaniec
Michał Oszmaniec
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Is there a HodgePoincare-Hopf Index theorem for non compact manifolds?

Does HodgePoincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomenon.

Is there a Hodge Index theorem for non compact manifolds?

Does Hodge index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomenon.

Is there a Poincare-Hopf Index theorem for non compact manifolds?

Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomenon.

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Michał Oszmaniec
Michał Oszmaniec
  • 965
  • 9
  • 23

Is there a Hodge Index theorem for a non compact manifoldmanifolds?

Does Hodge index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomenaphenomenon.

Is there a Hodge Index theorem for a non compact manifold?

Does Hodge index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomena.

Is there a Hodge Index theorem for non compact manifolds?

Does Hodge index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomenon.

Source Link
Michał Oszmaniec
Michał Oszmaniec
  • 965
  • 9
  • 23

Is there a Hodge Index theorem for a non compact manifold?

Does Hodge index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia).

Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomena.