Space Forms of Grassmann Manifolds

1963 ◽  
Vol 15 ◽  
pp. 193-205 ◽  
Author(s):  
Joseph A. Wolf
Keyword(s):  
Riemannian Manifolds ◽  
Grassmann Manifold ◽  
Space Form ◽  
Classification Problem ◽  
Grassmann Manifolds ◽  
Space Forms ◽  
Real Complex ◽  
Form Problem ◽  
Spherical Space Form ◽  
Real Euclidean Space

We shall consider the classification problem for space forms of (Riemannian manifolds which are covered by) real, complex, and quaternionic Grassmann manifolds. In the particular case of the real Grassmann manifold of oriented 1-dimensional subspaces of a real Euclidean space, this is the classical "spherical space form problem" of Clifford and Klein. We shall not consider space forms of the Cayley projective plane because it is easy to see that there are no non-trivial ones.

scholarly journals A class of C-totally real submanifolds of Sasakian space forms

1998 ◽  
Vol 65 (1) ◽  
pp. 120-128
Author(s):  
Filip Defever ◽  
Ion Mihai ◽  
Leopold Verstraelen
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Space Form ◽  
Sasakian Space Form ◽  
Real Submanifolds ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real ◽  
Real Complex

AbstractRecently, Chen defined an invariant δM of a Riemannian manifold M. Sharp inequalities for this Riemannian invariant were obtained for submanifolds in real, complex and Sasakian space forms, in terms of their mean curvature. In the present paper, we investigate certain C-totally real submanifolds of a Sasakian space form M2m+1(C)satisfying Chen's equality.

scholarly journals Topological spherical space form problem. III. Dimensional bounds and smoothing

1983 ◽  
Vol 106 (1) ◽  
pp. 135-143 ◽  
Author(s):  
Ib Madsen ◽  
Charles Thomas ◽  
C. Terence C. Wall
Keyword(s):  
Space Form ◽  
Spherical Space ◽  
Form Problem ◽  
Spherical Space Form

scholarly journals Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yanlin Li ◽  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Abimbola Abolarinwa ◽  
Rifaqat Ali
Keyword(s):  
Riemannian Manifolds ◽  
Space Form ◽  
Upper Bounds ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Sasakian Space Form ◽  
Slant Submanifold ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Sasakian Space Forms

This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the ϕ -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the ϕ -Laplacian operator on closed oriented m -dimensional slant submanifolds in a Sasakian space form M ~ 2 k + 1 ε is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the ϕ -Laplacian on slant submanifold in a sphere S 2 n + 1 with ε = 1 and ϕ = 2 .

scholarly journals The topological spherical space form problem—II existence of free actions

Topology ◽  
1976 ◽  
Vol 15 (4) ◽  
pp. 375-382 ◽  
Author(s):  
I. Madsen ◽  
C.B. Thomas ◽  
C.T.C. Wall
Keyword(s):  
Space Form ◽  
Spherical Space ◽  
Free Actions ◽  
Form Problem ◽  
Spherical Space Form

Metrics of Positive Scalar Curvature on Spherical Space Forms

1996 ◽  
Vol 48 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Boris Botvinnik ◽  
Peter B. Gilkey
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Eta Invariant ◽  
Positive Scalar ◽  
Spherical Space Forms ◽  
Simply Connected ◽  
Spherical Space Form

AbstractWe use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.

Triharmonic Riemannian submersions from 3-dimensional space forms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tomoya Miura ◽  
Shun Maeta
Keyword(s):  
Existence Theorem ◽  
Space Form ◽  
Dimensional Space ◽  
Riemannian Submersion ◽  
Partial Answer ◽  
Riemannian Submersions ◽  
Space Forms ◽  
3 Dimensional

Abstract We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.

scholarly journals Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

2021 ◽  
Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Riemannian Space ◽  
Space Forms ◽  
Type Formula ◽  
Isometric Embeddings ◽  
Curvature Measures ◽  
Riemannian Space Forms

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.

scholarly journals Lightlike Hypersurfaces of Indefinite Generalized Sasakian Space Forms

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Dae Ho Jin
Keyword(s):  
Vector Field ◽  
Space Form ◽  
Sasakian Manifold ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Characterization Theorems ◽  
Lightlike Hypersurfaces

We study lightlike hypersurfacesMof an indefinite generalized Sasakian space formM-(f1,f2,f3), with indefinite trans-Sasakian structure of type(α,β), subject to the condition that the structure vector field ofM-is tangent toM. First we study the general theory for lightlike hypersurfaces of indefinite trans-Sasakian manifold of type(α,β). Next we prove several characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form.

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