On invariant operations on pseudo-Riemannian manifolds

Jan Slovák

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 2, page 269-276
  • ISSN: 0010-2628

Abstract

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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].

How to cite

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Slovák, Jan. "On invariant operations on pseudo-Riemannian manifolds." Commentationes Mathematicae Universitatis Carolinae 33.2 (1992): 269-276. <http://eudml.org/doc/247392>.

@article{Slovák1992,
abstract = {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].},
author = {Slovák, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {invariant operators; natural operators; bundle functors; Chern forms; Pontrjagin forms; Chern form; Pontrjagin form; pseudo-Riemannian manifolds; natural operators},
language = {eng},
number = {2},
pages = {269-276},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On invariant operations on pseudo-Riemannian manifolds},
url = {http://eudml.org/doc/247392},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Slovák, Jan
TI - On invariant operations on pseudo-Riemannian manifolds
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 2
SP - 269
EP - 276
AB - 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].
LA - eng
KW - invariant operators; natural operators; bundle functors; Chern forms; Pontrjagin forms; Chern form; Pontrjagin form; pseudo-Riemannian manifolds; natural operators
UR - http://eudml.org/doc/247392
ER -

References

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  1. Atiyah M., Bott R., Patodi V.K., On the heat equation and the index theorem, Inventiones Math. 19 (1973), 279-330. (1973) Zbl0364.58016MR0650828
  2. Baston R.J., Eastwood M.G., Invariant operators, Twistors in mathematics and physics Lecture Notes in Mathematics 156 Cambridge University Press (1990). (1990) MR1089914
  3. Gilkey P.B., Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math. 10 (1973), 344-382. (1973) Zbl0259.58010MR0324731
  4. Gilkey P.B., Local invariants of a pseudo-Riemannian manifold, Math. Scand. 36 (1975), 109-130. (1975) Zbl0299.53040MR0375340
  5. Kolář I., Michor P.W., Slovák J., Natural operations in differential geometry, to appear in Springer-Verlag, 1992. MR1202431
  6. Nijenhuis A., Natural bundles and their general properties, in Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972 317-334. Zbl0246.53018MR0380862
  7. Slovák J., On invariant operations on a manifold with connection or metric, J. Diff. Geometry (1992), (to appear). (1992) MR1189498
  8. Stredder P., Natural differential operators on Riemannian manifolds and representations of the orthogonal and special orthogonal group, J. Diff. Geom. 10 (1975), 647-660. (1975) MR0415692
  9. Terng C.L., Natural vector bundles and natural differential operators, American J. of Math. 100 (1978), 775-828. (1978) Zbl0422.58001MR0509074
  10. Weyl H., The classical groups, Princeton University Press Princeton (1939). (1939) Zbl0020.20601MR1488158

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