This article is cited in 18 papers

Geometric Symbol Calculus\break for Pseudodifferential Operators. I

V. A. Sharafutdinov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: A connection on a manifold allows us to define the full symbol of a pseudodifferential operator in an invariant way. The latter is called the geometric symbol to distinguish it from the coordinate-wise symbol. The traditional calculus is developed for geometric symbols: an expression of the geometric symbol through the coordinate-wise symbol, formulas for the geometric symbol of the product of two operators, and of the dual operator.
The work consists of two parts. The first part considers operators on scalar functions. The second part generalizes main results to operators on vector bundles.

Key words: pseudodifferential operator, connection on a manifold, covariant derivative.

UDC: 517.98

Received: 09.07.2003

 English version: Siberian Advances in Mathematics, 2005, 15:3, 81–125

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