論文の公開元へGETZLER’S SYMBOL CALCULUS AND THE COMPOSITION OF DIFFERENTIAL OPERATORS ON CONTACT RIEMANNIAN MANIFOLDS
概要
operators by unifying two kinds of ideas: that of Widom ([14], [15]) about symbol calculus
on Riemannian manifold and that of Alvarez-Gaum´e ([1]) who used the Clifford variables
to propose a suitable filtration of symbol space. Getzler’s symbol calculus simplifies the
calculation of the principal part, or the top grading part (cf. [4], (2.10)), of the composite of
symbols ([7, Theorems 2.7 and 3.5]) and consequently provides a remarkably short proof of
the Atiyah-Singer index theorem for the Dirac operator ([7, §3], [8], [1]).
In this paper, on a contact Riemannian manifold with contact distribution H, following
Getzler’s idea we will introduce a similar symbol calculus of H-pseudodifferential operators,
which will turn out to be an effective tool for understanding the contact Riemannian structure from the viewpoint of calculus. The manifold possesses a canonical Spinc structure, the
Clifford variables associated with which provide similarly a filtration of symbol space, so
that Getzler’s idea can be applicable. The main result in this paper is Theorem 3.5, which
expressly offers an explicit formula for the top grading part of the composite of polynomial
symbols, that is, the symbols of H-differential operators. In the spin manifold case its counterpart is [7, Theorem 2.7], which was certified by using the Campbell-Hausdorff formula.
To prove Theorem 3.5, we will employ not the CH formula but the formula (1.1), which
gives an explicit expression of the connection coefficients of the hermitian Tanno connection. The CH formula is so daunting that Benameur-Heitsch [4], who applied Getzler’s idea
to the case of foliated spin manifold, used Atiyah-Bott-Patodi’s formula [2, Proposition 3.7]
for the proof of [4, Theorem 4.6] which is a foliation version of [7, Theorem 2.7]. Their idea
certainly led us to the application of (1.1), but the proof itself of the formula (3.10), which
is an essential part of Theorem 3.5, does not follow their strategy in [4, §4]. Our approach
2020 Mathematics Subject Classification. Primary 58J40; Secondary 53D35, 35S05.
The author was partially supported by JSPS KAKENHI Grant Number JP21K03219. ...
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