Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral Triples on Carnot Manifolds   We analyze whether one can construct a spectral triple for a Carnot manifold
$M$, which detects its Carnot-Carath\'{e}odory metric and its graded dimension.
Therefore we construct self-adjoint horizontal Dirac operators $D^H$ and show
that each horizontal Dirac operator detects the metric via Connes' formula, but
we also find that in no case these operators are hypoelliptic, which means they
fail to have a compact resolvent.
  First we consider an example on compact Carnot nilmanifolds in detail, where
we present a construction for a horizontal Dirac operator arising via pullback
from the Dirac operator on the torus. Following an approach by Christian B\"ar
to decompose the horizontal Clifford bundle, we detect that this operator has
an infinite dimensional kernel. But in spite of this, in the case of Heisenberg
nilmanifolds we will be able to discover the graded dimension from the
asymptotic behavior of the eigenvalues of this horizontal Dirac operator.
Afterwards we turn to the general case, showing that any horizontal Dirac
operator fails to be hypoelliptic. Doing this, we develop a criterion from
which hypoellipticity of certain graded differential operators can be excluded
by considering the situation on a Heisenberg manifold, for which a complete
characterization of hypoellipticity in known by the Rockland condition.
  Finally, we show how spectral triples can be constructed from horizontal
Laplacians via the Heisenberg pseudodifferential calculus developed by Richard
Beals and Peter Greiner. We suggest a few of these constructions, and discuss
under which assumptions it may be possible to get an equivalent metric to the
Carnot-Carath\'{e}odory metric from these operators. In addition, we mention a
formula by which the Carnot-Carath\'{e}odory metric can be detected from
arbitrary horizontal Laplacians.

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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-th