Geodesic
Geometric Symbol Calculus for Pseudodifferential Operators.~II
Matematičeskie trudy, Tome 8 (2005) no. 1, pp. 176-201.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 second part considers operators on vector bundles. 
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V. A. Sharafutdinov. Geometric Symbol Calculus for Pseudodifferential Operators.~II. Matematičeskie trudy, Tome 8 (2005) no. 1, pp. 176-201. https://geodesic-test.mathdoc.fr/item/MT_2005_8_1_a5/

[1] Sharafutdinov V. A., “Geometricheskoe ischislenie simvolov psevdodifferentsialnykh operatorov. I”, Mat. trudy, 7:2 (2004), 159–206 | MR