Submanifolds of constant sectional curvature in Pseudo-Riemannian manifolds

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds M ⁿ (K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\K \ne \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of R ⁿ which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n – 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n – 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\\overline M \] which have constant curvatures.