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However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities", say "curve of singularities", "surface of singularities", etc.

This method is applied in the following paper (and some of its references) which contains a counter example to an open conjecture on the number of limit cycles of Lienard equation.

https://www.ams.org/journals/proc/2007-135-06/S0002-9939-07-08688-1/S0002-9939-07-08688-1.pdf

So desingularisation of a manifold of singularities, instead of an isolated singularity, can have very strong consequences(giving a counter example to a conjecture or some other interesting consequences).

This situation is a motivation to ask the following question about an arbitrary Riemannian manifold $M$:

What would be some consequences and results of blowing up of the zero section, as the singular set of the geodesic flow on $TM$? Are there some papers or book devoting to this point of view?

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    $\begingroup$ Could you explain what you mean by "blowing up of the zero section, as the singular set of the geodesic flow on TM"? Blowing up is interesting only if the manifold is singular (for example, a cone) and you blow up the singularity, Your question appears to be about a smooth manifold without any singularities. Also, what role does the geodesic flow play in the the blow up? Note that the geodesic flow is also a smooth flow. It's just stationary along the zero section. $\endgroup$
    – Deane Yang
    Commented Sep 25, 2018 at 12:46
  • $\begingroup$ @DeaneYangMy apology for my delay. By blowing up I mean the blow-up of a singular point of a vector field: For an isolated singular point of a vector field $X$ at origin in $\mathbb{R}^n$, we define a map $p:S^{n-1}\times \mathbb{R}\to \mathbb{R}^n$ with $p(z,t)=tz$. Then we pull back the vector field X to a vector field on $S^{n-1}\times \mathbb{R}$. the resulting field vanishs at $S^{n-1}\times{0}$. Then we factor out a term $t^k$ for some $k$. This process is called blowing up of the isolated singularity at origin. THE word singular indicates that we $\endgroup$
    – Ali Taghavi
    Commented Sep 26, 2018 at 10:05
  • $\begingroup$ have a singularity of a vector field, not singularity of manifold. As a generalization of this process, in a paper by R. Roussarie and F. Dumortier, Memoir AMS(The refrence [1] of the linked paper , they apply the process for the case of a curve of singularity or a manifold of singularity. This methods leads to a blow up interpretation of the concept of CANARD.Now in geodesic flow, the zero section is a manifold of singularity for the geodesic flow. $\endgroup$
    – Ali Taghavi
    Commented Sep 26, 2018 at 10:10

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