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

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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. 
@article{MT_2005_8_1_a5,
     author = {V. A. Sharafutdinov},
     title = {Geometric {Symbol} {Calculus} for {Pseudodifferential} {Operators.~II}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {176--201},
     publisher = {mathdoc},
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     number = {1},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2005_8_1_a5/}
}
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V. A. Sharafutdinov. Geometric Symbol Calculus for Pseudodifferential Operators.~II. Matematičeskie trudy, Tome 8 (2005) no. 1, pp. 176-201. http://geodesic.mathdoc.fr/item/MT_2005_8_1_a5/