I would love to understand the famous formula $g\_{ij}(x) = \delta\_{ij} + \frac{1}{3}R\_{kijl}x^kx^l +O(\|x\|^3)$, which is valid in Riemannian normal coordinates and possibly more general situations.


I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's "Curvature in Mathematics and Physics" from which the question title and formula is stolen :-) or cf. S.Lang's "Differential and Riemannian Manifolds"]. The other proof involves computing that $\partial\_k\partial\_lg\_{ij}(x)$ shares some symmetries of curvature [cf. M.Spivak's "A Comprehensive Introduction to Differential Geometry, Vol. 2" where it is a several page "hairy computation" or cf. H.Weyl's 1923 edition of Riemann's Habilitationsvortrag (reprinted in a recent German book by Jürgen Jost) which I find uncomprehensible.]


**Are you aware of any other proof? Are normal coordinates necessary?**


While the Jacobi fields proof is short and elegant enough, it irks me that it requires "higher technology" not involved in the endproduct. Somehow the formula should be provable by pure calculus. Indeed, it is stated as an exercise in P.Petersen's "Riemannian Geometry": From the context I guess he thinks it should follow from the expression of $\partial\_lg\_{ij}$ as a sum of 2 Christoffel symbols and the simplified expression for curvature at $x=0$ where the Christoffel symbols vanish. Alas my attempts at this go in circles...


I find the situation quite amazing: Not many textbooks treat this fundamental and historic formula. (Estimating from the sample on my shelf it is $3/17.$ E.g. it seems it's not even in Levi-Civita's classic.)


**Update/Scholium:**


In classical language: The knackpoint seems to be a "differential Bianchi formula" for the Christoffel symbols at $0$. This follows from the geodesic equation. I see no other way yet.


A more modern approach minimizing (but not eliminating) the role of geodesics is in A.Gray's Tubes book. (Noted in comments. I'm waiting for [www.amazon.de](http://www.amazon.de) to deliver this treasure.)


$\bullet$ While geodesics are very geometric and normal coordinates are very practical, methinks the formula is a tad ungeometric. *What I'm hoping/asking for is a coordinate-independent formula for the second derivative of $g$ in terms of a suitable "reference connection".*



Riemann's formula for the metric in a normal neighborhood
My question has been answered in comments by Liviu Nicolaescu:


The (almost) ultimate proof (for my taste) is via A. Gray's formula(e) for (symmetric) higher covariant derivative(s) of normal coordinate vector fields. **Any exposition of normal coordinates lacking this formula is severely lacking.** (I'd prefer symmetrized c.d. of differentials of normal coordinates...). It is valid for any symmetric connection and gives Riemann's formula by application to some parallel bilinear form (e.g. the Riemann metric). (Riemann's really original Habilitationsvortrag formula needs a little additional algebra, cf. Dedekind/Weber [below] or Spivak.)


The second best approach is using Jacobi fields (cf. e.g. Le Spectre (LNM 194) for higher order terms without detail. (For hardcore syntacticists: Strook, Intro to An. of Paths on a Riem. Mf.)).


Third best: Repere mobile, but this seems Riemannian. (Cf. e.g. Heat Kernels and Dirac Operators, or Atiyah/Bott/Patodi appendix, or Cartan's Geometrie des Espaces de Riemann.)


The historically first proof is due to Dedekind/Weber (Anmerkung in "Bernhard Riemann's Gesammelte Mathematische Werke und Wissenschaftlicher Nachlaß", 1876/1892) via proving Riemann's Commentatio formula (which is also in Spivak) for curvature. (I could streamline this quite amazingly by doing the Levi-Civita in cotangent space.)


H.Weyl (1919/1923) has the historically second proof in his German edition of Riemann's Habilitationsvortrag, based on the extrinsic Gauss lemma (i.e. e.g. Besse, Einstein Mfs, Thrm. 1.45), which is what Spivak has worked out, and possibly has inspired the repere mobile proof (explicit in Heat Kernels [above]) - I don't (and am unwilling to) comprehend Cartan's proof.)


The nicest classical Riemannian proof is in L.P.Eisenhart, Riemannian Geometry, 1925/1949


I still don't have an opinion on R.Bryant's answer. It is possibly exactly what Riemann did for his inaugural lecture 1854. But then, Riemann had at least in later work (Commentatio, 1861) introduced Christoffel symbols (written $p\_{\iota,\iota^\prime,\iota^{\prime\prime}}$) and other formulas for curvature.


