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- W257079340 abstract "A submanifold M of a Riemannian manifold M is said to be parallel if the second fundamental form of % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca % WGnbaaaaaa!36D6! $$ overline M $$ is parallel. For example, an affine subspace M of IRm or a symmetric R-space M ∈ ℝm, which is minimally imbedded in a hypersphere of IRm (cf. Takeuchi-Kobayashi [12]), is a parallel submanifold of IRm. Ferus ([3], [4]) showed that essentially these submanifolds exhaust all parallel sub-manifolds of ℝm in the following sense: A complete full parallel submanifold of the Euclidean space IRm = Mm (0) is congruent to (a) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacqGH9a % qpcaWGjbGaamOuamaaCaaaleqabaGaamyBamaaBaaameaacaaIWaaa % beaaaaGccqGHxdaTcaWGnbWaaSbaaSqaaiaadYgaaeqaaOGaey41aq % RaaiOlaiaac6cacaGGUaGaamytamaaBaaaleaacaWGZbaabeaakiab % gkOimlaadMeacaWGsbWaaWbaaSqabeaacaWGTbWaaSbaaWqaaiaaic % daaeqaaaaakiabgwPiflaadMeacaWGsbWaaWbaaSqabeaacaWGTbWa % aSbaaWqaaiaaigdaaeqaaaaakiabgwPiflaac6cacaGGUaGaaiOlai % abgwPiflaadMeacaWGsbWaaWbaaSqabeaacaWGTbWaaSbaaWqaaiaa % dohaaeqaaaaakiabg2da9iaadMeacaWGsbWaaWbaaSqabeaacaWGTb % aaaOGaaiilaiaaysW7caWGTbGaeyypa0JaamyBamaaBaaaleaacaaI % WaaabeaakiabgUcaRmaaqaeabaGaamyBamaaBaaaleaacaWGPbaabe % aaaeqabeqdcqGHris5aOGaaiilaiaadohacqGHLjYScaaIWaGaaiil % aiaaysW7caWGVbGaamOCaiaaysW7caWG0bGaam4Baaaa!760C! $$M = I{R^{{m_0}}} times {M_l} times ...{M_s} subset I{R^{{m_0}}} oplus I{R^{{m_1}}} oplus ... oplus I{R^{{m_s}}} = I{R^m},;m = {m_0} + sum {{m_i}} ,s ge 0,;or;to$$ so ⩾ 0, or to (b) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacqGH9a % qpcaWGnbWaaSbaaSqaaiaadYgaaeqaaOGaey41aqRaaiOlaiaac6ca % caGGUaGaey41aqRaamytamaaBaaaleaacaWGZbaabeaakiabgkOiml % aadMeacaWGsbWaaWbaaSqabeaacaWGTbWaaSbaaWqaaiaadYgaaeqa % aaaakiabgwPiflaac6cacaGGUaGaaiOlaiabgwPiflaadMeacaWGsb % WaaWbaaSqabeaacaWGTbWaaSbaaWqaaiaadohaaeqaaaaakiabg2da % 9iaadMeacaWGsbWaaWbaaSqabeaacaWGTbaaaOGaaiilaiaad2gacq % GH9aqpdaaeabqaaiaad2gadaWgaaWcbaGaamyAaaqabaGccaGGSaGa % am4CaiabgwMiZkaaigdacaGGSaaaleqabeqdcqGHris5aaaa!6199! $$M = {M_l} times ... times {M_s} subset I{R^{{m_l}}} oplus ... oplus I{R^{{m_s}}} = I{R^m},m = sum {{m_i},s ge 1,}$$ where each Mi ⊂ ℝ is an irreducible symmetric R-space." @default.
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- W257079340 creator A5045478397 @default.
- W257079340 date "1981-01-01" @default.
- W257079340 modified "2023-10-13" @default.
- W257079340 title "Parallel Submanifolds of Space Forms" @default.
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- W257079340 doi "https://doi.org/10.1007/978-1-4612-5987-9_23" @default.
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