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Conformally Flat Manifolds Proc. Nat. Acad. Sci. USA Vol. 69, No. 9, pp. 2675-2676, September 1972 Conformally Flat Manifolds R. S. KULKARNI The Institute for Advanced Study, Princeton, New Jersey 08540 Communicated by D. C. Spencer, July 10, 1972 ABSTRACT This note describes some new examples of imbedding. We call a surgery on M' admissible if the cor- conformally flat manifolds, as a step toward a classification responding imbedding :SP X D'+1 wM' is admissible. We of such manifolds up to conformal equivalence. have the following A Riemannian manifold M' is called conformally flat if it THEOREM 1. Let M' be a conformally fiat manifold, and can be covered by neighborhoods { Ua. } such that there exists Ma a manifold obtained by performing admissible surgery on a conformal map s: Ua R'. From the point of view of con- Mln. Then ER admits a conformallyflat structure. formal geometry, conformally flat manifolds are the "core" We also have a similar manifolds. Primary examples of conformally flat manifolds are manifolds of constant curvature. Recall also that every 2- THEOREM 2. Let Mn, Nn be conformally flat manifolds. dimensional Riemannian manifold is conformally flat. This Then their connected sum admits a conformallyflat structure. follows from the existence of isothermal co-ordinates, and Hypersurfaces*. I have obtained a satisfactory classi- to of this fact is essentially equivalent the existence under- in > 4. A if fication of conformally flat hypersurfaces Rn+i, n lying complex structure the manifold is orientable. In this typical result is the following: way a conformally flat manifold may be considered as a generalization of a Riemann surface and one can expect THEOREM 3. Let Al" be a compact conformally flat manifold analogies between the theory of Riemann surfaces, complex n > 4 and t: 31- Rn+ a conformal immersion. Suppose that manifolds, and conformally flat manifolds. both the metric and immersion of A1 are analytic. Then Mn The notion of conformal flatness is one of the most primi- is diffeomorphic to Sn or Sn- X Si. tive concepts in differential geometry. In spite of this fact, The case when the metric and immersion are only as- excluding the case of a Riemann surface, most of the work sumed to be C' is more interesting. The basic local proposi- on conformally flat manifolds has been of local character. tion is the following: An exception is the work of Kuiper (3), who has notably ex- tended the idea of conformal development. One can pose PROPOSITION (Cartan, Schouten). A hypersurface in the following basic and very broad problem. Rn+iX n > 4, is conformally fiat if at least (n-i) eigenvalues of the secondfundamentalform are equal at each point. PROBLEM. Classify conformally flat manifolds up to con- Due to this proposition locally there turn out to be only formal equivalence. four possibilities: Considering the complexity of the theory of Riemann sur- (i) totally umbilic-hence of constant curvature; faces and that of space forms, it is obvious that the problem (ii) flat (but not necessarily totally umbilic); is extremely complicated. To start with, it is important to (iii) a surface of revolution: Let [t,x',. .,x8] be orthonormal have some classes of examples. It may be somewhat sur- co-ordinates in Rn+l. Choose a curve in the [t,x'] plane lying prising that all examples known so far are essentially the in the half plane xi > 0. Revolving this curve around the t- manifolds of constant curvature and certain of their products. axis, we obtained a surface of revolution. The purpose of this note is to describe some new examples. (iv) a tube: Choose any smooth curve in Rn+i and take its The details will be published elsewhere. normal sphere bundle where the spheres are of a sufficiently small fixed radius. By definition the total space of the bundle Conformal Surgery. I shall first describe a process is a tube. Roughly speaking a general conformally flat hy- of surgery on conformally flat manifolds. This shows in partic- persurface is obtained by smooth glueing of these four possi- ular that the class of conformally flat manifolds is very rich. bilities. However, arbitrary glueing is not possible. For ex- Recall that surgery on M' involves imbedding t:SP X DI+' ample a tube does not contain an umbilic so cannot be glued Mn, p + q + 1 = n, removing the interior of t(SP X Dq+l), to a totally umbilic piece. and smoothly glueing DP+' X SI along the boundary. A standard imbedding of SP X Dq+' in Rn, p + q + 1 = n, is the normal disc bundle of a standard SP lying in a (p + 1)-plane in * It came to my notice when I had already finished this work that Rn. An t:SP - is con- imbedding X D+ M¶ where Mn Cartan (1) and Schouten (4) also attempted classification of con- formally flat and p + q + 1 = n is called admissible if there formally flat hypersurfaces. However, both authors have ob- exists an isotopy tj of imbeddings 0 < t < 1 such that to = a, tained part of the local results only, and determination of global and such that there exists a neighborhood U of tl (SP X Dq+') form of, say, a complete hypersurface was not considered by and a conformal map so: U - Rn so that (o'ti is a standard them. 2675 Downloaded by guest on September 27, 2021 2676 Mathematics: Kulkarni Proc. Nat. Acad. Sci. USA 69 (1972) A notable consequence of this classification is formally analytically immersed as a complete hypersurface in Rn+l. THEOREM 4. Let M' be a conformaly flat manifold n > 4 Theorem 6 should be compared with the well-known result and :M - Rf+l a conformal immersion as a complete hy- persurface. Then Kr(Mn) is free. of Hilbert asserting nonexistence of a complete surface of COROLLARY. Let Mn be a compact space form (i.e., of constant negative curvature in RW. For the corresponding con- deeper results about conformal stant curvature) n > 4. If Mn $ Sin then Mn cannot be con- immersions of Riemann sur- formally immersed in Rn+1 faces see the works of Osserman, Garsia, Klotz, and Ruledy, In their recent work, Chern and Simons (2) have obtained see Riledy (5) and the references there. necessary conditions for conformal immersions of Riemannian I thank Profs. D. C. Spencer, M. F. Atiyah, and R. Bott, S. T. manifolds. These conditions are expressed in terms of secon- Yau and J. W. Wood for discussions that were useful in guiding my dary characteristic classes. The above corollary is not a thinking. Thanks to Professor Milnor for pointing out Ruedy's consequence of these conditions, e.g., for a flat torus Chern- paper. I also thank the Institute for Advanced Study and NSF Simons obstructions vanish but it cannot be conformally Grant GP20289, which provided the financial support. immersed as a hypersurface. On the other hand, when the method of Chern and Simons works, it will in general give 1. Cartan, E. (1917) "La deformation des hypersurfaces dans better results as it applies also for conformal immersions in l'espace conforme reel a n > 5 dimensions," Bull. Soc. Math. France, 45, 57-121, especially -23-28. arbitrary codimension. 2. Chern, S. S. & Simons, J. (1971) "Some Cohomology Classes The following are related results requiring a somewhat dif- in Principal Fiber Bundles," Proc. Nat. Acad. Sci. USA 68, ferent technique. 791-794. 3. Kuiper, N. H. (a) (1949) "On conformally-flat spaces in the THEOREM 5. (conformal rigidity of a cylinder). Let Mn - large," Ann. Math. 50, 916-924; (b) (1950) "On compact S' X Rn-i (product metric) n > 4 and t:Mn -Rn+ an conformally Euclidean spaces of dimension >2," Ann. analytic, conformal imbedding such that &(Mn) is complete. Math. 52, 478-490. 4. Schouten, J. A. (1921) "'Tber die konforme Abbildung. .," Then L(Mn) is again a flat cylinder and t is a homethety. Math. Zeit. 11, 58-88, especially §23. THEOREM 6. Let Ml be the complete, simply connected space 5. Ruedy, R. A. (1971) "Embeddings of open Riemann of constant negative curvature n > 4. Then Mn cannot be con- surfaces,"Comm. Math. Helv. 46, 214-225. Downloaded by guest on September 27, 2021.
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