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Zbl 0789.58076
Toth, G.
Operators on eigenmaps between spheres. (English)
[J] Compos. Math. 88, No.3, 317-332 (1993). ISSN 0010-437X; ISSN 1570-5846/e

As is known, the eigenspace ${\cal H}\sp p$ corresponding to the eigenvalue $\lambda\sb p = p(p+m-1)$ of the Laplacian $\Delta$ on $S\sp m$ consists of homogeneous harmonic polynomials and $m+1$ variables. A map $f: S\sp m \to S\sb V$ into the unit sphere of a Euclidean vector space $V$ is said to be a $\lambda\sb p$-eigenmap if all its components belong to $\cal H\sp p$. For $m = 3$, a full classification of all $\lambda\sb p$-eigenmaps was given by the author [Indiana Univ. Math. J. 36, 231-239 (1987; Zbl 0628.58010)].\par In the paper under review, it is proved that any nonzero $\text{SO}(m+1)$-module homomorphism $D$ of an orthogonal $\text{SO}(m+1)$-module $W$ into $({\cal H}\sp p)\sp* \otimes {\cal H}\sp q$ gives rise to an operator which carries $\lambda\sb p$-eigenmaps into $\lambda\sb q$-eigenmaps. Interesting examples are provided. Using Young tableaux, an explicit description of these operators for $W$ irreducible is given.\par As applications, the author studies the degree raising and lowering operators, infinitesimal rotations of eigenmaps and symmetrization.
[I.Mihai (Bucureşti)]

MSC 2000:: *58J50 Spectral problems; spectral geometry; scattering theory
53C20 Riemannian manifolds (global)

Keywords: eigenmap; $\text{SO}(m)$-module; Laplacian; Young tableaux

Citations: Zbl 0628.58010

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