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A characterization of the dependence of the Riemannian metric on the curvature tensor by Young symmetrizers. (English) Zbl 0904.53030

The first method for the determination of a Riemannian metric from the components of the curvature tensor in normal coordinates was given by Herglotz. The so-called Herglotz relations are nonlinear differential equations and give (at least in the analytic case) the power series expansion of the metric. The related coefficients are determined by the symmetrized partial derivatives of the curvature tensor. Later on, P. Günther found a power series expansion of the metric with explicit coefficients which depend on covariant derivatives of the curvature tensor symmetrized in a certain way.
In the paper under review, the author studies the difference between both kind of symmetrizations. For this reason the representation theory of the symmetric group \({\mathcal S}_r\) is used. The partial derivatives \(\partial_{i_1}\dots\partial_{i_r} R_{ijkl}\) induce group ring elements belonging to a direct sum \(J_{(r)}\oplus\widehat J_{(r)} \oplus \breve J_{(r)}\) of three minimal left ideals of the group ring \(\mathbb{C} [{\mathcal S}_{r+4}]\). Then the symmetrized partial derivatives \(\partial_{(i_1}\dots\partial_{i_r} R_{| a| i_{r+1} i_{r+2})b}\) correspond to a linear mapping of the direct sum into \(J_{ (r)} \cdot \varepsilon\), \(\varepsilon\in \mathbb{C} [{\mathcal S}_{r+4}]\) which maps \(\widehat J_{(r)} \oplus \breve J_{(r)}\) to 0. For the symmetrized covariant derivatives \(\nabla_{(i_1} \dots \nabla_{i_r)} R_{ijkl}\) and \(\nabla_{(i_1}\dots\nabla_{i_r} R_{| a| i_{r+1} i_{r+2})b}\) only the ideals \(J_{(r)}\) and \(J_{(r)} \cdot \varepsilon\) do appear. So, the inverse mapping \(J_{(r)} \cdot \varepsilon \to J_{(r)}\) gives a relation between both kinds of symmetrized covariant derivatives. This shows \({\mathcal R}= {\mathcal R}^s\), where \({\mathcal R}\) is the algebra of tensor polynomials generated by \(\nabla_{(i_1} \dots \nabla_{i_r)} R_{ijkl}\) and \({\mathcal R}^s\) is the algebra generated by \(\nabla_{(i_1} \dots \nabla_{i_r} R_{| k| i_{r+1} i_{r+2})l}\).
To prove these results, the author uses purely algebraic methods like Young symmetrizers and the Littlewood-Richardson rule. Furthermore, the author constructs examples of metrics for which the group ring elements induced by \(\partial^{(r)}R\) belong to \(J_{(r)}\) only or admit a nontrivial part in the ideal \(\widehat J_{(r)} \oplus \breve J_{(r)}\).

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
05E10 Combinatorial aspects of representation theory
53A55 Differential invariants (local theory), geometric objects

Software:

MathTensor; PERMS
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[1] Boerner, H.: Darstellungen von Gr-uppen (Die Grundlehren der mathematischen Wis- senschaften in Einzeldarstellungen: Vol. 74). Berlin - Göttingen - Heidelberg: Springer- Verlag, 1955.
[2] Boerner, H.: Representations of Groups (2. revised Ed.). Amsterdam: North-Holland Pubi. Comp., 1970. · Zbl 0112.26301
[3] Christensen, S. and L. Parker: MathTensor: A System for Doing Tensor Analysis by Computer. Reading (Mass.) et al.: Addison-Wesley, 1994.
[4] Fiedler, B.: PERMS 2.1. Leipzig: Mathematisches Institut, Universität Leipzig, 1997. Will be sended in to MathSource, Wolfram Research Inc.
[5] Fiedler, B.: A use of ideal decomposition in the computer algebra of tensor expressions. Z. Anal. Anw. 16 (1997), 145 - 164. · Zbl 0872.20017 · doi:10.4171/ZAA/756
[6] Fulling, S., King, R., Wybourne, B. and C. Cummins: Normal forms for tensor polyno- mials: I. The Riernann tensor. Class. Quantum Gray . 9 (1992), 1151 - 1197. · Zbl 0991.53517 · doi:10.1088/0264-9381/9/5/003
[7] Gunther, P.: Spinorkalkil und Normalkoordinaten. ZAMM 55 (1975), 203 - 210. · Zbl 0324.53012 · doi:10.1002/zamm.19750550502
[8] Gunther, P.: Huygens’ Principle and hyperbolic Equations (Perspectives in Mathematics: Vol. 5). Boston et al.: Academic Press, Inc., 1988. · Zbl 0655.35003
[9] Herglotz, G.: Ober die Bestimmung eines Linienelements in Normalkoordinaten aus dern Riemannschen Kriimmungstensor. Math. Annalen 93 (1925), 46 - 53.
[10] James, C. D. and A. Kerber: The Representation Theory of the Symmetric Group (Ency- clopedia of Mathematics and its Applications: Vol. 16). Reading (Mass.) et al.: Addison- Wesley Pubi. Comp., 1981. · Zbl 0491.20010
[11] Kerber, A.: Representations of Permutation Groups (Lecture Notes in Mathematics: Vol. 240, 495). Berlin - Heidelberg - New York: S p ringer- Verlag, 1971, 1975. · Zbl 0232.20014
[12] Kowalski, 0., Prüfer. F. and L. Vanhecke: D’Atri spaces. In: Topics in Geometry: In Memory of Joseph D’Atri. Ed.: Gindikin, S. Boston - Basel - Berlin: Birkhäuser, 1996, pp. 241 - 284. Reprint from: Progress in Nonlinear Differential Equations, Volume 20. · Zbl 0862.53039
[13] Littlewood, D.: The Theory of Group Characters and Matrix Representations of Groups (2. Ed.). Oxford: Clarendon Press, 1950. · Zbl 0038.16504
[14] Macdonald, I.: Symmetric Functsons and Hall Polynomials. Oxford: Clarendon Press, 1979. · Zbl 0487.20007
[15] Naimark, M. and A. stern: Theory of Group Representations (Grundlehren der Mathe- matischen Wissenschaften: Vol. 246). Berlin - Heidelberg - New York: Springer-Verlag, 1982.
[16] van der Waerden, B.: Algebra. 9., 6. Ed., Vol. I, II. Berlin et at.: Springer-Verlag, 1993. · Zbl 0781.12002
[17] Weyl, II.: The Classical Groups, their Invariants and Representations. Princeton (New Jersey): Princeton University Press, 1939. · Zbl 0020.20601
[18] Wolfram, S.: Mathematica (2. Ed.). Bonn et al.: Addison-Wesley, 1992. In German.
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