Geodesic
The laplace operator on normal homogeneous Riemannian manifolds
Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 3-40.

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The article presents an information about the Laplace operator defined on the real-valued mappings of compact Riemannian manifolds, and its spectrum; some properties of the latter are studied. The relationship between the spectra of two Riemannian manifolds connected by a Riemannian submersion with totally geodesic fibers is established. We specify a method of calculating the spectrum of the Laplacian for simply connected simple compact Lie groups with biinvariant Riemannian metrics, by representations of their Lie algebras. As an illustration, the spectrum of the Laplacian on the group $\operatorname{SU}(2)$ is found. 
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V. N. Berestovskii; V. M. Svirkin. The laplace operator on normal homogeneous Riemannian manifolds. Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 3-40. http://geodesic.mathdoc.fr/item/MT_2009_12_2_a0/

[1] Adams D., Lektsii po gruppam Li, Nauka, M., 1979 | MR | Zbl

[2] Bers L., Dzhon F., Shekhter M., Uravneniya s chastnymi proizvodnymi, Mir, M., 1966 | MR | Zbl

[3] Besse A., Mnogoobraziya s zamknutymi geodezicheskimi, Mir, M., 1981 | MR

[4] Veil G., Klassicheskie gruppy, ikh invarianty i predstavleniya, Izd-vo inostr. lit., M., 1947

[5] Veil A., Integrirovanie v topologicheskikh gruppakh i ego primenenie, Izd-vo inostr. lit., M., 1950

[6] Vladimirov V. S., Zharinov V. V., Uravneniya matematicheskoi fiziki, Fizmatlit, M., 2000

[7] Diksme Zh., Universalnye obertyvayuschie algebry, Mir, M., 1978 | MR

[8] Zhelobenko D. P., Kompaktnye gruppy Li i ikh predstavleniya, Nauka, M., 1970 | MR | Zbl

[9] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[10] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, T. 1, 2, M., 1981

[11] Onischik A. L., Topologiya tranzitivnykh grupp preobrazovanii, Fizmatlit, M., 1995 | MR

[12] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964

[13] Berger M., “Geometry of the spectrum. I”, Differential Geometry, Part 2 (Stanford Univ., Stanford, Calif., 1973), Proc. Sympos. Pure Math., 27, Amer. Math. Soc., Providence, RI, 1975, 129–152 | MR

[14] Berger M., Gauduchon P., Mazet E., Le Spectre d'une Vari'et'e Riemannienne, Lecture Notes in Math., 194, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[15] D'Atri J. E., Nickerson H. K., “The existence of special orthonormal frames”, J. Differential Geometry, 2 (1968), 393–409 | MR

[16] Duitstermaat J. J., Guillemin V. W., “Spectral geometry of real and complex manifolds”, Differential Geometry, Part 2 (Stanford Univ., Stanford, Calif., 1973), Proc. Sympos. Pure Math., 27, Amer. Math. Soc., Providence, RI, 1975, 205–209 | MR

[17] Fegan H. D., “The spectrum of the Laplacian on forms over a Lie group”, Pacific J. Math., 90:2 (1980), 373–387 | MR | Zbl

[18] Gilkey P. B., “The spectral geometry of real and complex manifolds”, Differential Geometry, Part 2 (Stanford Univ., Stanford, Calif., 1973), Proc. Sympos. Pure Math., 27, Amer. Math. Soc., Providence, RI, 1975, 265–280 | MR

[19] Milnor J., “Eigenvalues of the Laplace operator on certain manifolds”, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542 | DOI | MR | Zbl

[20] Onishchik A. L., Lectures on Real Semisimple Lie Algebras and Their Representations, ESI Lectures in Mathematics and Physics, European Math. Soc. (EMS), Zurich, Switzerland, 2004 | MR | Zbl

[21] Rosenberg S., The Laplacian on a Riemannian Manifold, An Introduction to Analysis on Manifolds, London Mathematical Society Student Texts, 31, Cambridge University Press, Cambridge, 1997 | MR | Zbl