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CR Embedded Submanifolds of CR Manifolds
About this Title
Sean N. Curry, Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand and A. Rod Gover, Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 258, Number 1241
ISBNs: 978-1-4704-3544-8 (print); 978-1-4704-5073-1 (online)
DOI: https://doi.org/10.1090/memo/1241
Published electronically: March 11, 2019
Keywords: CR embeddings,
CR invariants,
tractor calculus,
Gauss-Codazzi-Ricci equations,
Bonnet theorem.
MSC: Primary 32V05, 53B15; Secondary 32V30, 53B25, 53A30
Table of Contents
Chapters
- 1. Introduction
- 2. Weighted Tanaka-Webster Calculus
- 3. CR Tractor Calculus
- 4. CR Embedded Submanifolds and Contact Forms
- 5. CR Embedded Submanifolds and Tractors
- 6. Higher Codimension Embeddings
- 7. Invariants of CR Embedded Submanifolds
- 8. A CR Bonnet Theorem
Abstract
We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. We define a normal tractor bundle in the ambient standard tractor bundle along the submanifold and show that the orthogonal complement of this bundle is not canonically isomorphic to the standard tractor bundle of the submanifold. By determining the subtle relationship between submanifold and ambient CR density bundles we are able to invariantly relate these two tractor bundles, and hence to invariantly relate the normal Cartan connections of the submanifold and ambient manifold by a tractor analogue of the Gauss formula. This leads also to CR analogues of the Gauss, Codazzi, and Ricci equations. The tractor Gauss formula includes two basic invariants of a CR embedding which, along with the submanifold and ambient curvatures, capture the jet data of the structure of a CR embedding. These objects therefore form the basic building blocks for the construction of local invariants of the embedding. From this basis we develop a broad calculus for the construction of the invariants and invariant differential operators of CR embedded submanifolds.
The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naïve methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.
- M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. MR 650828, DOI 10.1007/BF01425417
- T. N. Bailey, M. G. Eastwood, and A. R. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), no. 4, 1191–1217. MR 1322223, DOI 10.1216/rmjm/1181072333
- Toby N. Bailey, Michael G. Eastwood, and C. Robin Graham, Invariant theory for conformal and CR geometry, Ann. of Math. (2) 139 (1994), no. 3, 491–552. MR 1283869, DOI 10.2307/2118571
- Eric Bedford, Proper holomorphic mappings, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 157–175. MR 733691, DOI 10.1090/S0273-0979-1984-15235-2
- Francis E. Burstall and David M. J. Calderbank, Submanifold geometry in generalized flag manifolds, Rend. Circ. Mat. Palermo (2) Suppl. 72 (2004), 13–41. MR 2069394
- F. E. Burstall & D. M. J. Calderbank, Conformal submanifold geometry I-III, preprint, (2010), arXiv:1006.5700v1 [math.DG].
- David M. J. Calderbank and Tammo Diemer, Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67–103. MR 1856258, DOI 10.1515/crll.2001.059
- Andreas Čap and A. Rod Gover, Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), no. 4, 1511–1548. MR 1873017, DOI 10.1090/S0002-9947-01-02909-9
- Andreas Čap and A. Rod Gover, CR-tractors and the Fefferman space, Indiana Univ. Math. J. 57 (2008), no. 5, 2519–2570. MR 2463976, DOI 10.1512/iumj.2008.57.3359
- Andreas Čap and Jan Slovák, Parabolic geometries. I, Mathematical Surveys and Monographs, vol. 154, American Mathematical Society, Providence, RI, 2009. Background and general theory. MR 2532439
- Andreas Čap, Jan Slovák, and Vladimír Souček, Bernstein-Gelfand-Gelfand sequences, Ann. of Math. (2) 154 (2001), no. 1, 97–113. MR 1847589, DOI 10.2307/3062111
- E. Cartan, Sur la géometrié pseudo-conforme des hypersurfaces de deux variables complexes. I, Ann. Math. Pura Appl. 11 (1932) 17-90; II, Ann. Scoula Norm. Sup. Pisa 1 (1932) 333–354.
- S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146
- Sean N. Curry and A. Rod Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, Asymptotic analysis in general relativity, London Math. Soc. Lecture Note Ser., vol. 443, Cambridge Univ. Press, Cambridge, 2018, pp. 86–170. MR 3792084
- Sorin Dragomir, On pseudo-Hermitian immersions between strictly pseudoconvex CR manifolds, Amer. J. Math. 117 (1995), no. 1, 169–202. MR 1314462, DOI 10.2307/2375040
- Sorin Dragomir and André Minor, CR immersions and Lorentzian geometry: Part I: Pseudohermitian rigidity of CR immersions, Ric. Mat. 62 (2013), no. 2, 229–263. MR 3120051, DOI 10.1007/s11587-013-0157-5
- Sorin Dragomir and Giuseppe Tomassini, Differential geometry and analysis on CR manifolds, Progress in Mathematics, vol. 246, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2214654
- Michael Eastwood, Notes on conformal differential geometry, The Proceedings of the 15th Winter School “Geometry and Physics” (Srní, 1995), 1996, pp. 57–76. MR 1463509
- Michael G. Eastwood and John W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. 109 (1987), no. 2, 207–228. MR 880414
- Peter Ebenfelt, Xiaojun Huang, and Dmitri Zaitsev, Rigidity of CR-immersions into spheres, Comm. Anal. Geom. 12 (2004), no. 3, 631–670. MR 2128606, DOI 10.4310/CAG.2004.v12.n3.a6
- James J. Faran V, A reflection principle for proper holomorphic mappings and geometric invariants, Math. Z. 203 (1990), no. 3, 363–377. MR 1038706, DOI 10.1007/BF02570744
- Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
- Charles Fefferman, Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), no. 2, 131–262. MR 526424, DOI 10.1016/0001-8708(79)90025-2
- Charles Fefferman and C. Robin Graham, Conformal invariants, Astérisque Numéro Hors Série (1985), 95–116. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837196
- Charles Fefferman and C. Robin Graham, The ambient metric, Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. MR 2858236
- A. Rod Gover, Invariants on projective space, J. Amer. Math. Soc. 7 (1994), no. 1, 145–158. MR 1214703, DOI 10.1090/S0894-0347-1994-1214703-8
- A. Rod Gover, Invariant theory for a parabolic subgroup of $\textrm {SL}(n+1,\textbf {R})$, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1543–1553. MR 1231035, DOI 10.1090/S0002-9939-1995-1231035-5
- A. Rod Gover, Invariants and calculus for projective geometries, Math. Ann. 306 (1996), no. 3, 513–538. MR 1415076, DOI 10.1007/BF01445263
- A. Rod Gover, Aspects of parabolic invariant theory, Rend. Circ. Mat. Palermo (2) Suppl. 59 (1999), 25–47. The 18th Winter School “Geometry and Physics” (Srní, 1998). MR 1692257
- A. Rod Gover, Invariant theory and calculus for conformal geometries, Adv. Math. 163 (2001), no. 2, 206–257. MR 1864834, DOI 10.1006/aima.2001.1999
- A. Rod Gover, Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature, J. Geom. Phys. 60 (2010), no. 2, 182–204. MR 2587388, DOI 10.1016/j.geomphys.2009.09.016
- A. Rod Gover and C. Robin Graham, CR invariant powers of the sub-Laplacian, J. Reine Angew. Math. 583 (2005), 1–27. MR 2146851, DOI 10.1515/crll.2005.2005.583.1
- A. R. Gover and A. Waldron, Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem, preprint, (2015), arXiv:1506.02723 [math.DG].
- D. H. Grant, A conformally invariant third order Neumann-type operator for hypersurfaces, MSc thesis, University of Auckland, 2003.
- Kengo Hirachi, Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), no. 1, 151–191. MR 1745015, DOI 10.2307/121115
- Bernhard Lamel, A reflection principle for real-analytic submanifolds of complex spaces, J. Geom. Anal. 11 (2001), no. 4, 627–633. MR 1861300, DOI 10.1007/BF02930759
- John M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), no. 1, 411–429. MR 837820, DOI 10.1090/S0002-9947-1986-0837820-2
- Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155–158. MR 88629, DOI 10.2307/1970121
- Michael J. Markowitz, CR-hypersurfaces in a space with a pseudoconformal connection, Trans. Amer. Math. Soc. 276 (1983), no. 1, 117–132. MR 684496, DOI 10.1090/S0002-9947-1983-0684496-0
- A. Minor, CR embeddings, chains, and the Fefferman bundle, preprint, (2013), arXiv:1010.3458v3 [math.DG].
- H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo 23 (1907), 185–220.
- R. Stafford, Tractor calculus and invariants for conformal submanifolds, MSc thesis, University of Auckland, 2006.
- Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of $n$ complex variables, J. Math. Soc. Japan 14 (1962), 397–429. MR 145555, DOI 10.2969/jmsj/01440397
- Noboru Tanaka, Graded Lie algebras and geometric structures, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 147–150. MR 0222802
- Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book-Store Co., Ltd., Tokyo, 1975. Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9. MR 0399517
- Noboru Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131–190. MR 589931, DOI 10.4099/math1924.2.131
- Yoshiya Takemura, On the invariant submanifold of a CR-manifold, Kodai Math. J. 5 (1982), no. 3, 416–425. MR 684799
- Y. Vyatkin, Manufacturing conformal invariants of hypersurfaces, PhD thesis, The University of Auckland, 2013.
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599
- S. M. Webster, On mapping an $n$-ball into an $(n+1)$-ball in complex spaces, Pacific J. Math. 81 (1979), no. 1, 267–272. MR 543749
- S. M. Webster, The rigidity of C-R hypersurfaces in a sphere, Indiana Univ. Math. J. 28 (1979), no. 3, 405–416. MR 529673, DOI 10.1512/iumj.1979.28.28027