Curvature operators and characteristic classes

Author:
Irl Bivens

Journal:
Trans. Amer. Math. Soc. **269** (1982), 301-310

MSC:
Primary 53C21; Secondary 57R20

DOI:
https://doi.org/10.1090/S0002-9947-1982-0637040-7

MathSciNet review:
637040

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given tensors and of type on a Riemannian manifold we construct in a natural way a form . If and satisfy the generalized Codazzi equations then this form is closed. In particular if denotes the th curvature operator then is (up to a constant multiple) the th Pontrjagin class of . By means of a theorem of Gilkey we give conditions sufficient to guarantee that a form constructed from more complicated expressions involving the curvature operators does in fact belong to the Pontrjagin algebra. As a corollary we obtain Thorpe's vanishing theorem for manifolds with constant th sectional curvature.

If at each point in the tangent space contains a subspace of a particular type (similar to curvature nullity) we show that certain Pontrjagin classes must vanish. We generalize the result that submanifolds of Euclidean space with flat normal bundle have a trivial Pontrjagin algebra.

The curvature operator, , is interesting in that the components of with respect to any orthonormal frame are given by certain universal (independent of frame) homogeneous linear polynomials in the components of the curvature tensor. We characterize all such operators and using this characterization derive in a natural way the Weyl component of .

**[1]**M. Atiyah, R. Bott and V. K. Patodi,*On the heat equation and the index theorem*, Invent. Math.**19**(1973), 279-330. MR**0650828 (58:31287)****[2]**S. S. Chern,*On the curvature and characteristic classes of a Riemannian manifold*, Abh. Math. Sem. Univ. Hamburg**20**(1956), 117-126. MR**0075647 (17:783e)****[3]**S. S. Chern and N. H. Kuiper,*Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space*, Ann. of Math. (2)**56**(1952), 422-430. MR**0050962 (14:408e)****[4]**Y. K. Cheung and C. C. Hsiung,*Curvature and characteristic classes of compact Riemannian manifolds*, J. Differential Geom.**1**(1967), 89-97. MR**0217738 (36:827)****[5]**L. P. Eisenhart,*Riemannian geometry*, Princeton Univ. Press, Princeton, N. J., 1949. MR**0035081 (11:687g)****[6]**A. Gray,*Some relations between curvature and characteristic classes*, Math. Ann.**184**(1969), 257-267. MR**0261492 (41:6105)****[7]**R. S. Kulkarni,*On the Bianchi identities*, Math. Ann.**199**(1972), 175-204. MR**0339004 (49:3767)****[8]**I. M. Singer and J. A. Thorpe,*The curvature of*-*dimensional Einstein spaces*, Global Analysis, Papers in Honor of K. Kodaira (D. C. Spencer and S. Iyanaga, editors), Princeton Univ. Press, Princeton, N. J., 1969, pp. 355-365. MR**0256303 (41:959)****[9]**A. Stehney,*Courbure d'ordre p et les classes de Pontrjagin*, J. Differential Geom.**8**(1973), 125-133. MR**0362333 (50:14775)****[10]**J. Thorpe,*Sectional curvature and characteristic classes*, Ann. of Math. (2)**80**(1964), 429-443. MR**0170308 (30:546)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C21,
57R20

Retrieve articles in all journals with MSC: 53C21, 57R20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0637040-7

Article copyright:
© Copyright 1982
American Mathematical Society