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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 $$(1, m)$$}.

##### MSC:
 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
##### Keywords:
Gauss-Bonnet formula; Finsler spaces; Chern connection
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