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A note on the Gauss-Bonnet theorem for Finsler spaces. (English) Zbl 0849.53046
With regard to generalizations of the Gauss-Bonnet formula to Finsler manifolds there exists some literature [A. Lichnerowicz, Comment. Math. Helv. 22, 271-301 (1949; Zbl 0039.17501); P. Dazord, Bull. Soc. Math. Fr. 99, 171-192 (1971; Zbl 0215.23504); ibid. 99, 397 (1971; Zbl 0222.53059); H. Rund, Monatsh. Math. 79, 233-252 (1975; Zbl 0298.53024); the reviewer, Tensor, New Ser. 49, No. 1, 9-17 (1990; Zbl 0724.53013); the reviewer, Differential geometry and its applications, Proc. Conf., Dubrovnik/Yugosl. 1988, 179-197 (1989; Zbl 0681.53015)].
In the present paper the formula is successfully generalized to Finsler spaces which have constant \(\text{Vol(Finsler } S^{n- 1})= \text{Vol}(I_x M, h_x)\) (the Riemannian volume of the indicatrix \(I_x M\)). The two-dimensional case goes as follows: Let \((M, F)\) be a compact, connected Finsler surface without boundary, where \(F\) is the fundamental function. Suppose the function \(\text{Vol}(x)\) has the constant value \(\text{Vol(Finsler } S^1)\), then the value of \(\int_M= \int_M V^*(- \Omega^2_1/2\pi)\) is independent of any vector field \(V\) on \(M\) and \(\int_M= \chi(M)\{\text{Vol(Finsler } S^1)/\text{Vol}(S^1)\}\), where \(\Omega^2_1= {{R_1}^2}_{12} \omega^1\wedge \omega^2+ P_{2111} \omega^1 \wedge \omega^2_1\) is one of the curvature 2-forms of the Chern connection [the authors, Houston J. Math. 19, No. 1, 135-180 (1993; Zbl 0787.53018)].
{Reviewer’s remark: The authors refer to the Chern connection. It seems, however, to the reviewer that \(P_{ijkl}\) and \(R_{ijkl}\) coincide respectively with the \(hv\)- and \(h\)-curvature tensors \(F_{ijkl}\) and \(K_{ijkl}\) of the Rund connection [the reviewer, Foundations of Finsler geometry and special Finsler spaces (1986; Zbl 0594.53001)]. In the two-dimensional case we have \({{K_1}^2}_{12}= {{K_h}^i}_{jk} l^h m_i l^j m^k= R\) (\(h\)-scalar-curvature of the Cartan connection) in the Berwald frame \((1, m)\)}.

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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