Sentence examples for be a harmonic map from inspiring English sources

Let be a harmonic map between compact manifolds and.

Let be a harmonic map, a vector field along, and a one-parameter variation of.

Moreover, taking the -sphere or the complex projective plane, letting be a harmonic map, and a Jacobi field, isotropy to first order is immediately guaranteed.

Let ({mathbb{D}}={z: |z|<1}) denote the unit disk, (w z)) be a harmonic mapping defined in ({mathbb{D}}).

Let (f z)=u z)+iv z)) be a harmonic mapping from (mathbb {H}) onto itself and continuous on (mathbb {H}cup mathbb {R}) with (f infty)=infty).

(1) Assume that (f : M rightarrow N) is a harmonic map such that, (n : = dim_{{mathbb {C}}} N), ( H_{2n}(f, {mathbb {Z}}) ne 0).

Assume that (f : M rightarrow N) is a harmonic map such that, (exists i ge 2k) such that ( H_{i}(f, {mathbb {Z}}) ne 0).

(I) Assume that (f : M rightarrow N) is a harmonic map between two compact Kähler manifolds and that the curvature tensor of N is strongly negative.

Assume that (f : M rightarrow N) is a harmonic map such that, (n : = dim_{{mathbb {C}}} N), ( H_{2n}(f, {mathbb {Z}}) ne 0).

The first variation of the energy function vanishes precisely when f is a harmonic map, i.e., (Delta (f) = 0), where (Delta (f) : = Tr (nabla (Df)),) (nabla ) being the connection on (TM^{vee } otimes f^* (TN)) induced by the Levi-Civita connections on M and N. The energy functional enters also in the study of geodesics and Morse theory (see [288]).

If we additionally suppose that f is a harmonic mapping, then there exists a holomorphic function (g:mathbb {H}to mathbb {C}) such that (f z)= operatorname {Re}g z) +icy).