Simplified connectivity, then we further give such isoparametric hypersurface a rigidity
Simplified connectivity, then we further give such isoparametric hypersurface a rigidity classi( n -2) c fication. Under the assumption 0 P n and primarily based on [29], Mn should be isometric n to Hk ( a) Sn-k (b) S1 1 (c), exactly where k 1, n – 1, a 0, b 0 and 1 1 = 1 ; its a c b principal curvatures are provided by = c – a and = c – b (53)| |with multiplicities k and n – k, respectively. So, with each other with c = H, ||two and P are provided, respectively, by H= and P=(n – k) kc n-1 c , | |2 = -nn (54)1 k ( k – 1) c2 ( n – 2) c – (n – k)(n – k – 1) . n n ( n – 1) (55)Right here, the final two equations hold due to the equality k (n – k) = n – 1 with k 1, n – 1. Since 0 c, from (55), we’ve n-2 c – 0, n-2 c , n n n -2 c – nc2 P=when k = 1,- , 0 , when k = n – 1.(56)Therefore, we conclude that k = 1 simply because of 0 P n-2 c and Mn is often a hyperbolic n cylinder H1 ( a) Sn-1 (b). It can be quick to verify ||2 = ( P, n, c) when substituting (56) in to the second equation of (54). With each other with (53), (56) and 1 1 = 1 , we lastly get a c b b= nP ncP , a= , n-2 nP – (n – 2)c (57)Mathematics 2021, 9,14 ofand complete the proof of Theorem 1. Proof of Theorem 3. Considering the fact that tr(3 ) = 0, i.e., (19) holds for k = n , then Lemma 2, with each other two with Lemma 1, implies that(nH )1 ||two (n – 2)||two n(n – 1) P . n-Using Lemma 4 and following the proof of Theorem 2, Mn is often a completely umbilical n hypersurface and, by (18), it really is entirely geodesic if and only if P = c. In certain, if L1 1 is actually a geodesically comprehensive simply-connected Einstein manifold, applying precisely the same procedure as within the proof of Theorem 1 or Theorem 2, we obtain that such completely umbilical hypersurface have to be a sphere Sn ( R) and it is a completely geodesic sphere Sn (c) if and only if P = c. This completes the proof of Theorem three. 7. Conclusions Within this paper, we investigate the spacelike hypersurface immersed in Lorentz manifolds. One usually solves this trouble by using the Bochner approach combined using the maximum principle. Right here, with some suitable capabilities, we extend the ambient manifold to a far more generalized Ricci Benidipine manufacturer symmetric manifold; then, we receive some rigidity classifications when the ambient manifold is definitely an Einstein manifold. These capabilities are also applicable to (spacelike) submanifolds in (pseudo) Riemannian manifolds, which means that several final results of your isometric immersion theory of submanifolds is usually generalized. Meanwhile, we give several non-trivial examples in an effort to prove the existence in the Ricci symmetric manifolds satisfying the curvature situations (1) and (2). The Okumura-type inequality (19) introduced in [26] also implies the case of hypersurfaces with two distinct principal curvatures. Even so, we weren’t able to point out whether or not this inequality has a specific geometric significance.Author Contributions: Conceptualization, J.L.; formal evaluation, X.X., J.L. and C.Y.; writing–review, J.L.; methodology, J.L., X.X. and C.Y.; writing–original draft, X.X. All authors have study and agreed towards the published version with the manuscript. Funding: This Diversity Library custom synthesis function was supported by the National All-natural Science Foundation of China (Grant Nos. 12161078, 11761061) along with the Foundation for Distinguished Young Scholars of Gansu Province (Grant No. 20JR5RA515). Acknowledgments: The authors are extremely grateful towards the associate editor and referees for their helpful comments and recommendations. Institutional Assessment Board Statement: Not applicable. Informed Consent Statement: Not applicable. Information Availability Statement: Not applicable. Conflicts o.
