Ref Request: Geometric Du-Bois Raymond Lemma

Question

I've been looking everywhere for a reference for this "geometric" Du Bois-Raymond Lemma:

Here $\frac{DY}{dt}$ denotes the covariant derivative along $y$.

Answer 1

I found the second reference you gave in your comment, see [3]. In this paper, the authors refer to

the following “Du Bois-Raymond lemma”, a straightforward extension of the classical lemma to a Riemannian manifold

Apart from the misspell, it seems that du Bois-Reymond's classical lemma has its own wikipedia article, which includes bibliographic references. Note that this wikipedia entry is in French, and the English version does not seem to have as much information. Du Bois-Reymond's original papers were published in [1, 2].

[1] P. du Bois-Reymond, Erläuterungen zu den Anfangsgründen der Variationsrechnung », Mathematische Annalen 15, no 2,‎ 1879, p. 283-314

[2] P. du Bois-Reymond, Fortsetzung der Erläuterungen zu den Anfangsgründen der Variationsrechnung, Mathematische Annalen 15, no 2,‎ 1879, p. 564-576

[3] J. Langer, and D.A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology 24 (1985), no. 1, 75--88.