SPECTRAL RIGIDITY AND INVARIANT DISTRIBUTIONS ON ANOSOV SURFACES
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Authors
Paternain, Gabriel P
Salo, Mikko
Uhlmann, Gunther
Publication Date
2014Journal Title
JOURNAL OF DIFFERENTIAL GEOMETRY
ISSN
0022-040X
Publisher
International Press of Boston
Volume
98
Issue
1
Pages
147-181
Type
Article
Metadata
Show full item recordCitation
Paternain, G. P., Salo, M., & Uhlmann, G. (2014). SPECTRAL RIGIDITY AND INVARIANT DISTRIBUTIONS ON ANOSOV SURFACES. JOURNAL OF DIFFERENTIAL GEOMETRY, 98 (1), 147-181. https://doi.org/10.4310/jdg/1406137697
Abstract
This article considers inverse problems on closed Riemannian surfaces whose
geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and
injectivity of the geodesic ray transform on solenoidal 2-tensors. We also
establish surjectivity results for the adjoint of the geodesic ray transform on
solenoidal tensors. The surjectivity results are of independent interest and
imply the existence of many geometric invariant distributions on the unit
sphere bundle. In particular, we show that on any Anosov surface $(M,g)$, given
a smooth function $f$ on $M$ there is a distribution in the Sobolev space
$H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to
$M$ is the given function $f$.
Keywords
math.DG, math.DG, math.AP, math.DS
Identifiers
External DOI: https://doi.org/10.4310/jdg/1406137697
This record's URL: https://www.repository.cam.ac.uk/handle/1810/284378
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