# The Atiyah-Singer index formula for subelliptic operators on contact manifolds /

Abstract (Summary)

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic
differential operator. The topological index depends on a cohomology class that is constructed
from the principal symbol of the operator.
On contact manifolds, the naturally arising geometric operators are not elliptic, but subelliptic.
A filtration on the algebra of differential operators that is adapted to these geometric
structures, naturally leads to a symbolic calculus that is noncommutative, and a corresponding
subelliptic theory can be developed.
For such subelliptic operators we construct a symbol class in the K-theory of a noncommutative
C?-algebra naturally associated to the algebra of symbols. There is a canonical map from
this noncommutative K-theory to the ordinary cohomology of the manifold, which gives a class
to which the Atiyah-Singer formula can be applied. In this way we define the topological index
of a subelliptic operator, and we prove that it is equal to its analytic index.
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Advisor:

School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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