riemannian space forms
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Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.



Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1175
Author(s):  
Ion Mihai ◽  
Radu-Ioan Mihai

We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.



2021 ◽  
Vol 112 (1) ◽  
Author(s):  
Huili Liu ◽  
Yixuan Liu


Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 7
Author(s):  
Ion Mihai ◽  
Radu-Ioan Mihai

We give a simple proof of the Chen inequality for the Chen invariant δ(2,⋯,2)︸kterms of submanifolds in Riemannian space forms.





2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.



Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 395
Author(s):  
Jinhua Qian ◽  
Xueshan Fu ◽  
Seoung Dal Jung

In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are defined. The relationships of such partner curves are revealed including the relationship of their Frenet frames and the curvatures. Furthermore, the Darboux associated curves of k-type null helices are characterized and the conclusion that a null curve and its self-associated curve share the same Darboux associated curve is obtained.



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