scholarly journals Harmonic maps and totally geodesic maps between metric spaces

2004 ◽  
Vol 28 (28) ◽  
pp. 1-69
Author(s):  
Shin-ichi OHTA
Keyword(s):  
Metric Spaces ◽  
Harmonic Maps ◽  
Totally Geodesic

scholarly journals UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES

2002 ◽  
Vol 04 (04) ◽  
pp. 725-750 ◽  
Author(s):  
CHIKAKO MESE
Keyword(s):  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Harmonic Maps ◽  
Uniqueness Theorems ◽  
Singular Spaces ◽  
Domain Space ◽  
Recent Developments

Recent developments extend much of the known theory of classical harmonic maps between smooth Riemannian manifolds to the case when the target is a metric space of curvature bounded from above. In particular, the existence and regularity theorems for harmonic maps into these singular spaces have been successfully generalized. Furthermore, the uniqueness of harmonic maps is known when the domain has a boundary (with a smallness of image condition if the target curvature is bounded from above by a positive number). In this paper, we will address the question of uniqueness when the domain space is without a boundary in two cases: one, when the curvature of the target is strictly negative and two, for a map between surfaces with nonpositive target curvature.

Totally geodesic maps into metric spaces

2003 ◽  
Vol 244 (1) ◽  
pp. 47-65 ◽  
Author(s):  
S.-i. Ohta
Keyword(s):  
Metric Spaces ◽  
Totally Geodesic

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

2005 ◽  
Vol 16 (09) ◽  
pp. 1017-1031 ◽  
Author(s):  
QUN HE ◽  
YI-BING SHEN
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Second Variation ◽  
Finsler Manifolds ◽  
Totally Geodesic ◽  
The Stability ◽  
The Second Variation ◽  
Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

scholarly journals The second variation formula for exponentially harmonic maps

1999 ◽  
Vol 59 (3) ◽  
pp. 509-514 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Compact Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Sharp Contrast ◽  
Second Variation ◽  
Variation Formula ◽  
Totally Geodesic ◽  
The Second Variation

We derive the formula in the title and deduce some consequences. For example we show that the identity map from any compact manifold to itself is always stable as an exponentially harmonic map. This is in sharp contrast to the harmonic or p-harmonic cases where many such identity maps are unstable. We also prove that an isometric and totally geodesic immersion of Sm into Sn is an unstable exponentially harmonic map if m ≠ n and is a stable exponentially harmonic map if m = n.

SLANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

2012 ◽  
Vol 10 (02) ◽  
pp. 1250080 ◽  
Author(s):  
BAYRAM ṢAHIN
Keyword(s):  
Projective Space ◽  
Riemannian Manifolds ◽  
Complex Projective Space ◽  
Harmonic Maps ◽  
Sufficient Conditions ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Necessary And Sufficient ◽  
Complex Projective

We introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant immersions, invariant Riemannian maps and anti-invariant Riemannian maps. We give examples, obtain characterizations and investigate the harmonicity of such maps. We also obtain necessary and sufficient conditions for slant Riemannian maps to be totally geodesic. Moreover, we relate the notion of slant Riemannian maps to the notion of pseudo horizontally weakly conformal (PHWC) maps which are useful for proving various complex-analytic properties of stable harmonic maps from complex projective space.

scholarly journals Exponentially Harmonic Maps into Spheres

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 88
Author(s):  
Sorin Dragomir ◽  
Francesco Esposito
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Codimension Two ◽  
Geodesic Submanifold

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m ⊂ R m + 1 . Given a codimension two totally geodesic submanifold Σ ⊂ S m , we show that every nonconstant exponentially harmonic map ϕ : M → S m either meets or links Σ . If H 1 ( M , Z ) = 0 then ϕ ( M ) ∩ Σ ≠ ∅ .

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