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 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.

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 Submanifolds and the length of the second fundamental tensor

1983 ◽  
Vol 34 (1) ◽  
pp. 1-6
Author(s):  
Thomas Hasanis
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Mean Curvature ◽  
Sufficient Condition ◽  
Fundamental Tensor ◽  
The Mean

AbstractA sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

ON FINSLER MANIFOLDS WHOSE TANGENT BUNDLE HAS THE g-NATURAL METRIC

2011 ◽  
Vol 08 (07) ◽  
pp. 1593-1610 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Tangent Bundle ◽  
Contact Structure ◽  
Jacobi Operator ◽  
Riemannian Metric ◽  
Finsler Manifolds ◽  
Almost Contact Structure ◽  
Flat Riemannian Manifold

In this paper, we introduce a Riemannian metric [Formula: see text] and a family of framed f-structures on the slit tangent bundle [Formula: see text] of a Finsler manifold Fn = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fn is of positive constant curvature 1. Finally, we prove that (i) Fn is a locally flat Riemannian manifold if and only if [Formula: see text], (ii) the Jacobi operator [Formula: see text] is zero or commuting if and only if (M, F) have the zero flag curvature.

On Finsler surfaces of constant curvature with two-dimensional isometry group

2015 ◽  
Vol 26 (07) ◽  
pp. 1550046
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Constant Curvature ◽  
Isometry Group ◽  
Flag Curvature ◽  
New Technique ◽  
Finsler Metrics ◽  
Two Dimensional ◽  
Constant Flag Curvature ◽  
A New Technique

In this paper, we study Finsler surfaces of constant (flag) curvature. We show that the space of those, with two-dimensional isometric group depends on two arbitrary constants. We also give a new technique to recover Finsler metrics from the specified two constants. Using this technique we obtain some new Finsler surfaces of constant flag curvature with two-dimensional isometry group.

scholarly journals ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

2011 ◽  
Vol 08 (03) ◽  
pp. 501-510 ◽  
Author(s):  
HAMID REZA SALIMI MOGHADDAM
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Lie Groups ◽  
Three Dimensional ◽  
Flag Curvature ◽  
Nilpotent Lie Groups ◽  
Randers Metric ◽  
Levi Civita Connection ◽  
Simply Connected ◽  
Randers Metrics

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

scholarly journals A Characterization of Invariant Affine Connections

1960 ◽  
Vol 16 ◽  
pp. 35-50 ◽  
Author(s):  
Bertram Kostant
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Theorem ◽  
Transitive Group ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Simply Connected

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

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