By Zhongmin Shen

This e-book is a entire file of contemporary advancements in Finsler geometry and Spray geometry. Riemannian geometry and pseudo-Riemannian geometry are handled because the detailed case of Finsler geometry. The geometric tools constructed during this topic are beneficial for learning a few difficulties coming up from biology, physics, and different fields.

Audience: The e-book may be of curiosity to graduate scholars and mathematicians in geometry who desire to transcend the Riemannian global. Scientists in nature sciences will locate the geometric equipment offered helpful.

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**Extra info for Differential geometry of spray and Finsler spaces**

**Example text**

Non-isomorphic) axn-degeneracy graphs with U nodes. Proof (1) There exists no 2x4-degeneracy graph with 10 nodes (cf. Appendix, C3 and Tab. C3)1). 1) Appendix C4 contains a further counter-example: There exists no 2xS-degeneracy graph with 13 nodes (cf. Tab. CS). (2) There are two non-isomorphic 2x4 degeneracy graphs with 12 nodes (cf. Appendix, C3 and Tab. C3)1). Moreover, the following results from Appendix C: The nodes (or degeneracy tableaux) of 2xn-degeneracy graphs (n fixed) may be divided into classes of types such that each type of node (or type of tableau) can uniquely be assigned to a 2xn-degeneracy graph.

Proof (1) There exists no 2x4-degeneracy graph with 10 nodes (cf. Appendix, C3 and Tab. C3)1). 1) Appendix C4 contains a further counter-example: There exists no 2xS-degeneracy graph with 13 nodes (cf. Tab. CS). (2) There are two non-isomorphic 2x4 degeneracy graphs with 12 nodes (cf. Appendix, C3 and Tab. C3)1). Moreover, the following results from Appendix C: The nodes (or degeneracy tableaux) of 2xn-degeneracy graphs (n fixed) may be divided into classes of types such that each type of node (or type of tableau) can uniquely be assigned to a 2xn-degeneracy graph.

8 In the case n = (J there is only one basic form of arrangement for mini- mally laid degeneracy tableaux, since a "double diagonal form" can 1) For example both tableaux in Fig. 2 (3) are equivalent in form. However, in Fig. 2 (1) all tableaux are different in form. In Fig. 2 (2) two tableaux are equivalent in form while the form of the third is different. 43 always be obtained by an appropriate column exchange (cf. Fig. 2 (3)). Therefore,n =a fixed, all axa-degeneracy tableaux are equivalent.