Theorem 1, together with the author's extensions of the Sturmian Comparison Theorems, will suffice to establish the basic Theorem 27.3, p. 211 of Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold. Mathematical Notes, Princeton University Press, Princeton, N.J., 1976. Here and in the preceding reference there is given a Weierstrass Integral on a compact, connected, Riemannian manifold Mn, conditioned as in the above reference. Let γ be an arbitrary extremal of J joining a point P to a point Q, ≠ P on Mn. An extremal ζ of J joining A to B is termed nondegenerate if A and B are not conjugate on ζ. The index of ζ is by definition the “count” of conjugate points of A on ζ definitely preceding B.THEOREM 1. When the extremal γ is nondegenerate, extremals of J issuing from P with the same length as γ and sufficiently near γ in the sense of Fréchet, are nondegenerate and have the same index as γ.
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机译：定理1以及作者对Sturmian比较定理的扩展，足以建立基本定理27.3，p。全局变分分析211：Weierstrass在黎曼流形上积分。数学注释，普林斯顿大学出版社，新泽西州普林斯顿，1976年。在此和前面的参考文献中，在紧紧相连的黎曼流形Mn上给出了Weierstrass积分，其条件与上述参考文献相同。设γ是将点P连接到点Q的J的任意极值，Mn上的≠P。如果A和B不共轭在ζ上，则将A与B相连的J的极值ζ称为非简并的。根据定义，ζ的索引是肯定在 B 之前的ζ上A的共轭点的“计数”。定理1。长度与γ相同，在Fréchet的意义上足够接近γ，它们是不退化的，并且具有与γ相同的折射率。
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