Spectrum of the Lichnerowicz Laplacians on the spheres and the real projective spaces
Let \((M,g)\) be a smooth Riemannian manifold, \(S^p M\) the space of symmetric forms of degree \(p\) on \(M\) and \(\Delta^p _{(M,g)} : S^p M \rightarrow S^p M\) the Lichnerowicz Laplacian, \(p\in \mathbb{N} \) [see \textit{A. Lichnerowicz}, Inst. Hautes Études Sci. Publ. Math. 10, 293-344 (1961; Zbl 0098.42607)]. \( \Delta^0 _{(M,g)} : C^\infty (M) \rightarrow C^\infty (M) \) (resp. \( \Delta^1 _{(M,g)} : \Omega^1 (M) \rightarrow \Omega^1 (M) \)) coincides with the Laplace-Beltrami operator (resp. Hodge-de Rham Laplacian), where \( \Omega^1 (M) \) denotes the space of smooth differential \( 1-\)forms on \( M. \) Let \( S^n \) be the standard sphere endowed with the canonical Riemannian metric \( g _{can} .\) Using the decompositions \( S^2 S^n = \text{Ker } \delta_1 \oplus \delta^\star_1 (\Omega^1 (S^n)) = \text{Ker } \delta_1 \cap \text{Tr}^{-1} (0) \oplus (\delta_1^\star (\Omega^1 (S^n)) + C^\infty (S^n) g _{can}), \) where \( \delta_1^\star : \Omega^1 (S^n) \rightarrow S^2 S^n \) is defined by \( \delta_1^\star (\omega))= L _{\# \omega} (g _{can}), \;\delta_1 : S^2 S^n \rightarrow \Omega^1 (S^n) \) is the formal adjoint of \( \delta_1^\star \) and \( \text{Tr} : S^2 S^n \rightarrow C^\infty (S^n) \) the trace with respect to \( g _{can} \) [see \textit{M. Berger} and \textit{D. Ebin}, J. Differ. Geom. 3, 379-392 (1969; Zbl 0194.53103); \textit{A. Besse}, Einstein manifolds (1987; Zbl 0613.53001)rbrack{}, the author determines explicitly the spectrum and the eigenspaces of \( \Delta^2 _{(S^n, g_{can})} . \) As consequences, some classical results are pointed out: the Berger-Ebin conformal representation on \( (S^2, g _{can}) \) [see loc. cit.] and the rigidity of the canonical Einstein structure on \( (S^n , g _{can}) \) [see \textit{A. Besse}, loc. cit.]. In particular, the spectra and the eigenspaces of \( \Delta^p _{(\mathbb{R} P^n, g_{can})}\), \(p=1,2, \) are derived.
scientific article; zbMATH DE number 1418167