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invariant operators; natural operators; bundle functors; Chern forms; Pontrjagin forms
Invariant polynomial operators on Riemannian manifolds are well understood and the knowledge of full lists of them becomes an effective tool in Riemannian geometry, [Atiyah, Bott, Patodi, 73] is a very good example. The present short paper is in fact a continuation of [Slovák, 92] where the classification problem is reconsidered under very mild assumptions and still complete classification results are derived even in some non-linear situations. Therefore, we neither repeat the detailed exposition of the whole setting and the technical tools, nor we include all details of the proofs, the interested reader can find them in the above paper (or in the monograph [Kolář, Michor, Slovák]). After a short introduction, we study operators homogeneous in weight on oriented pseudo-Riemannian manifolds. In particular, we are interested in those of weight zero. The results involve generalizations of some well known theorems by [Gilkey, 75] and [Stredder, 75].
Atiyah M., Bott R., Patodi V.K.: On the heat equation and the index theorem. Inventiones Math. 19 (1973), 279-330. MR 0650828 | Zbl 0364.58016
Baston R.J., Eastwood M.G.: Invariant operators. Twistors in mathematics and physics Lecture Notes in Mathematics 156 Cambridge University Press (1990). MR 1089914
Gilkey P.B.: Curvature and the eigenvalues of the Laplacian for elliptic complexes. Advances in Math. 10 (1973), 344-382. MR 0324731 | Zbl 0259.58010
Gilkey P.B.: Local invariants of a pseudo-Riemannian manifold. Math. Scand. 36 (1975), 109-130. MR 0375340 | Zbl 0299.53040
Kolář I., Michor P.W., Slovák J.: Natural operations in differential geometry. to appear in Springer-Verlag, 1992. MR 1202431
Nijenhuis A.: Natural bundles and their general properties. in Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972 317-334. MR 0380862 | Zbl 0246.53018
Slovák J.: On invariant operations on a manifold with connection or metric. J. Diff. Geometry (1992), (to appear). MR 1189498
Stredder P.: Natural differential operators on Riemannian manifolds and representations of the orthogonal and special orthogonal group. J. Diff. Geom. 10 (1975), 647-660. MR 0415692
Terng C.L.: Natural vector bundles and natural differential operators. American J. of Math. 100 (1978), 775-828. MR 0509074 | Zbl 0422.58001
Weyl H.: The classical groups. Princeton University Press Princeton (1939). MR 1488158 | Zbl 0020.20601
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