Fiber structures of special (4 + 3 + 1) warped-like manifolds with Spin(7) holonomy

2014 ◽  
Vol 11 (08) ◽  
pp. 1450076
Author(s):  
Selman Uğuz ◽  
İbrahim Ünal
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Warped Product ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Simply Connected ◽  
Block Diagonal

A generalization of 8-dimensional multiply-warped product manifolds is considered as a special warped product, by allowing the fiber metric to be non-block diagonal. Motivating from the previous paper [S. Uğuz and A. H. Bilge, (3 + 3 + 2) warped-like product manifolds with Spin(7) holonomy, J. Geom. Phys.61 (2011) 1093–1103], we present a special warped product as a (4 + 3 + 1) warped-like manifold of the form M = F × B, where the base B is a 1-dimensional Riemannian manifold, and the fiber F is of the form F = F1 × F2 where Fi's (i = 1, 2) are Riemannian 4- and 3-manifolds, respectively. It is showed that the connection on M is entirely determined provided that the Bonan 4-form is closed. Assuming that the Fi's are complete, connected and simply connected, it is proved that the 3-dimensional fiber is isometric to S3 with constant curvature k > 0. Finally, the geometric properties of the 4-dimensional fiber of M are studied.

Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

2004 ◽  
Vol 70 (2) ◽  
pp. 301-319 ◽  
Author(s):  
Y. Nikolayevsky
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Curvature Operator ◽  
Specific Function ◽  
3 Dimensional ◽  
Space Of Constant Curvature

A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

scholarly journals Randers manifolds of positive constant curvature

2003 ◽  
Vol 2003 (18) ◽  
pp. 1155-1165 ◽  
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Complete Riemannian Manifold ◽  
Flag Curvature ◽  
Dimensional Sphere ◽  
Randers Metric ◽  
Constant Flag Curvature ◽  
Simply Connected

We prove that any simply connected and complete Riemannian manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it.

scholarly journals Doubly Warped Product Submanifolds of a Riemannian Manifold of Nearly Quasi-constant Curvature

2021 ◽  
Vol 53 ◽  
Author(s):  
Mehraj Ahmad Lone ◽  
Mohamd Saleem Lone ◽  
Mohammad Hasan Shahid
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Warped Product ◽  
Sharp Inequality

In the present paper, we form a sharp inequality for a doubly warped product submanifold of a Riemannian manifold of nearly quasi-constant curvature.

scholarly journals Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

2006 ◽  
Vol 58 (2) ◽  
pp. 282-311 ◽  
Author(s):  
M. E. Fels ◽  
A. G. Renner
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Isotropy Subgroup ◽  
Algebraic Classification ◽  
Ricci Flat ◽  
Einstein Spaces ◽  
Simply Connected

AbstractA method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ℝ4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

scholarly journals Doubly Warped Product Submanifolds of a Riemannian Manifold of Quasi-Constant Curvature

2015 ◽  
Vol 61 (1) ◽  
pp. 235-244
Author(s):  
Sibel Sular
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Warped Product ◽  
Sharp Inequality

Abstract We establish a sharp inequality for a doubly warped product submanifold in a Riemannian manifold of quasi-constant curvature.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

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