Invariants of locally conformally flat manifolds

Authors:
Thomas Branson, Peter Gilkey and Juha Pohjanpelto

Journal:
Trans. Amer. Math. Soc. **347** (1995), 939-953

MSC:
Primary 53C25; Secondary 53A30, 57R15, 58G26

DOI:
https://doi.org/10.1090/S0002-9947-1995-1282884-3

MathSciNet review:
1282884

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Abstract: Let be a locally conformally flat manifold with metric . Choose a local coordinate system on so where is the Euclidean standard metric. A polynomial in the derivatives of with coefficients depending smoothly on is a local invariant for locally conformally flat structures if the expression is independent of the choice of . Form valued local invariants are defined similarly. In this paper, we study the properties of the associated de Rham complex. We show that any invariant form can be obtained from the previously studied local invariants of Riemannian structures by restriction. We show the cohomology of the de Rham complex of local invariants is trivial. We also obtain the following characterization of the Euler class. Suppose that for an invariant polynomial , the integral vanishes for any locally conformally flat metric on the torus . Then up to the divergence of an invariantly defined one form, the polynomial is a constant multiple of the Euler integrand.

**[1]**I. Anderson,*The variational bicomplex*, Academic Press, Boston, MA (to appear). MR**1188434 (94a:58045)****[2]**A. Avez,*Characteristic classes and Weyl tensor. applications to general relativity*, Proc. Nat. Acad. Sci. U.S.A.**66**(1970), 265-268. MR**0263098 (41:7703)****[3]**T. Branson,*Sharp inequalities, the functional determinant, and the complementary series*, Trans. Amer. Math. Soc. (to appear). MR**1316845 (96e:58162)****[4]**T. Branson, S.-Y. A. Chang, and P. Yang,*Estimates and extremals for zeta function determinants on four manifolds*, Comm. Math. Phys.**149**(1992), 241-262. MR**1186028 (93m:58116)****[5]**T. Branson and B. Orsted,*Conformal indices of Riemannian manifolds*, Compositio Math.**60**(1986), 261-293. MR**869104 (88b:58131)****[6]**-,*Conformal geometry and global invariants*, Differential Geom. Appl.**1**(1991), 279-308. MR**1244447 (94k:58154)****[7]**P. Gilkey,*Local invariants of an embedded Riemannian manifold*, Ann. of Math.**102**(1975), 187-203. MR**0394693 (52:15492)****[8]**-,*Leading terms in the asymptotics of the heat equation*, Contemp. Math.**73**(1988), 79-85. MR**954631 (89h:58199)****[9]**-,*Invariance theory, the heat equation, and the Atiyah Singer index theorem*, 2nd ed., CRC Press (to appear). MR**1396308 (98b:58156)****[10]**S. Goldberg,*Curvature and homology*, Pure Appl. Math., vol. 11, Academic Press, 1962. MR**0139098 (25:2537)****[11]**J. Lee and T. Parker,*The Yamabe problem*, Bull. Amer. Math. Soc. (N.S.)**17**(1987), 37-91. MR**888880 (88f:53001)****[12]**E. Miller, Ph.D. thesis, M.I.T.**[13]**P. J. Olver,*Conservation laws and null divergences*, Math. Proc. Cambridge Philos. Soc.**94**(1983), 529-540. MR**720804 (85a:58092)****[14]**E. Onofri,*On the positivity of the effective action in a theory of random surfaces*, Comm. Math. Phys.**86**(1982), 321-326. MR**677001 (84j:58043)****[15]**H. Weyl,*The classical groups*, Princeton Univ. Press, Princeton, NJ, 1946.

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1282884-3

Article copyright:
© Copyright 1995
American Mathematical Society