Abstract
In this work, we show an injectivity result and support theorems for integral moments of a symmetric m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of symmetric m-tensor fields were first introduced by Sharafutdinov. First we generalize a Helgason type support theorem proven by Krishnan and Stefanov (Inverse Probl Imaging 3(3):453–464, 2009). We use this extended result along with the first integral moments of a symmetric m-tensor field to prove the aforementioned results.
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Acknowledgements
We would like to thank Prof. Vladimir Sharafutdinov for suggesting this problem. Besides, we would also like to express our sincere gratitude to Prof. Todd Quinto and Prof. Venky Krishnan for several hours of fruitful discussions. The authors benefited from the support of the Airbus Group Corporate Foundation Chair “Mathematics of Complex Systems” established at TIFR Centre for Applicable Mathematics and TIFR International Centre for Theoretical Sciences, Bangalore, India. Finally, we would like to thank the referee for their helpful comments and suggestions.
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Communicated by Alex Iosevich.
Appendix
Appendix
Proof of Lemma 1
First, let us recall that for a symmetric \((m-1)\)-tensor field v,
The idea here is to use an inductive argument for \(0 \le k \le m\). We start by showing the result for \(k=0,1,\) and then for general \(k\le m\).
From this, we see that the result is true for \(k=0\) and 1. Now, we are going to prove that the result is also true for \(k\le m\). Consider
where
Now, we will simplyfy each of the above terms one by one. Consider
After putting the values of \( J, J^1_k\) and \(J_k^2\) in \(\mathrm {d}v\), we get
\(\square \)
Proof of Estimate (11)
Let \(^{t}L= \frac{\Phi _{\xi }\partial _{\xi }}{i\lambda |\Phi _\xi |^{2}}\). Then as already noted
Consider,
Using the fact that, f is compactly supported and using (7), we get (11). \(\square \)
Proof of the Estimate (13)
Consider
Rewrite the above as :
Using [18, Remark 2.10], we get
Lemma 8
is a formal analytic symbol.
Proof
Let,
Then from the Cauchy integral formula [18, Exercise 2.4],
Hence,
is a formal analytic symbol \({{B}}^{i_{1}\ldots i_{m}}(x,y,\eta ;\lambda )\) by [18, Exercise 1.1]. \(\square \)
Hence,
But,
since,
see [18, Remark 2.10]. So, this along with (10) and (11), gives us:
\(\square \)
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Abhishek, A., Mishra, R.K. Support Theorems and an Injectivity Result for Integral Moments of a Symmetric m-Tensor Field. J Fourier Anal Appl 25, 1487–1512 (2019). https://doi.org/10.1007/s00041-018-09649-7
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DOI: https://doi.org/10.1007/s00041-018-09649-7