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reflections; (locally) symmetric immersions; extrinsic (locally) symmetric submanifolds; parallel immersions; (locally) symmetric spaces
We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also taken into consideration and several examples are given.
[BR] E. Backes and H. Reckziegel: On symmetric submanifolds of spaces of constant curvature. Math. Ann. 263 (1983), 419–433. DOI 10.1007/BF01457052 | MR 0707240
[C] B. Y. Chen: Geometry of submanifolds. Pure and Appl. Math. 22, Marcel Dekker, New York, 1973. MR 0353212 | Zbl 0262.53036
[CO] B. Y. Chen and K. Ogiue: On totally real submanifolds. Trans. Amer. Math. Soc. 193 (1974), 257–266. DOI 10.1090/S0002-9947-1974-0346708-7 | MR 0346708
[ChV] B. Y. Chen and L. Vanhecke: Isometric, holomorphic and symplectic reflections. Geom. Dedicata 29 (1989), 259–277. DOI 10.1007/BF00572443 | MR 0995302
[F] D. Ferus: Symmetric submanifolds of Euclidean space. Math. Ann. 247 (1980), 81–93. DOI 10.1007/BF01359868 | MR 0565140 | Zbl 0446.53041
[KN] S. Kobayashi and K. Nomizu: Foundations of differential geometry, I, II. Interscience Publ., New York, 1963, 1969. MR 0152974
[K] M. Kon: On some complex submanifolds in Kaehler manifolds. Canad. J. Math. 26 (1974), 1442–1449. DOI 10.4153/CJM-1974-138-5 | MR 0380658 | Zbl 0297.53013
[KV] O. Kowalski and L. Vanhecke: Geodesic spheres and a new recursion formula on Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino 45 (1987), 119–132. MR 0981158
[N2] H. Naitoh: Isotropic submanifolds with parallel second fundamental forms in symmetric spaces. Osaka J. Math. 17 (1980), 95–110. MR 0558321 | Zbl 0427.53022
[N1] H. Naitoh: Totally real parallel submanifolds in $P^{n}(c)$. Tokyo J. Math. 4 (1981), 279–306. DOI 10.3836/tjm/1270215155 | MR 0646040
[N] H. Naitoh: Parallel submanifolds of complex space forms I. Nagoya Math. J. 90 (1983), 85–117. DOI 10.1017/S0027763000020365 | MR 0702254 | Zbl 0509.53046
[N3] H. Naitoh: Symmetric submanifolds of compact symmetric spaces. Differential Geometry of Submanifolds, Proceedings, Kyoto 1984, K. Kenmotsu (ed.), Lecture Notes in Math. 1090, Springer-Verlag, Berlin, Heidelberg, New York, 1984, pp. 116–128. MR 0775150 | Zbl 0546.53034
[NT1] H. Naitoh and M. Takeuchi: Totally real submanifolds and symmetric bounded domains. Osaka J. Math. 19 (1982), 717–731. MR 0687769
[NT] H. Nakagawa and R. Takagi: On locally symmetric Kaehler submanifolds in a complex projective space. J. Math. Soc. Japan 28 (1976), 638–667. DOI 10.2969/jmsj/02840638 | MR 0417463
[NV] L. Nicolodi and L. Vanhecke: Rotations on a Riemannian manifold. Proc. Workshop on Recent Topics in Differential Geometry, Puerto de La Cruz 1990, D. Chinea and J. M. Sierra (eds.), Secret. Public. Univ. de La Laguna, Serie Informes 32, 1991, pp. 89–101. MR 1127455
[Nomi] K. Nomizu: Conditions for constancy of the holomorphic sectional curvature. J. Differential Geom. 8 (1973), 335–339. DOI 10.4310/jdg/1214431649 | MR 0380690 | Zbl 0279.53031
[SV] K. Sekigawa and L. Vanhecke: Symplectic geodesic symmetries on Kähler manifolds. Quart. J. Math. Oxford 37 (1986), 95–103. DOI 10.1093/qmath/37.1.95 | MR 0830633
[S] W. Strübing: Symmetric submanifolds of Riemannian manifolds. Math. Ann. 245 (1979), 37–44. DOI 10.1007/BF01420428 | MR 0552577
[T] M. Takeuchi: Parallel submanifolds of space forms. Manifolds and Lie groups, Papers in honor of Yozô Matsushima eds J. Hano, A. Morimoto, S. Murakami, K. Okamoto, H. Ozeki, Progress in Math., Birkhäuser, Boston, Basel, Stuttgart, 1981, pp. 429–447. MR 0642871 | Zbl 0481.53047
[TV] Ph. Tondeur and L. Vanhecke: Reflections in submanifolds. Geom. Dedicata 28 (1988), 77–85. DOI 10.1007/BF00147801 | MR 0965832
[Ts] K. Tsukada: Parallel Kaehler submanifolds of Hermitian symmetric spaces. Math. Z. 190 (1985), 129–150. DOI 10.1007/BF01159170 | MR 0793355 | Zbl 0568.53031
[V] L. Vanhecke: Geometry in normal and tubular neighborhoods. Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al vol. 58 (1988), 73–176. MR 1122858
[VW] L. Vanhecke and T. J. Wilmore: Interactions of tubes and spheres. Math. Ann. 263 (1983), 31–42. DOI 10.1007/BF01457081 | MR 0697328
[YM] K. Yano and I. Mogi: On real representations of Kaehlerian manifolds. Ann. of Math. 61 (1955), 170–189. DOI 10.2307/1969627 | MR 0068291
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