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I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates. The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},\dots$ at the point.

It is easy to get using Jacobi equation, but I would prefer to have a reference (if it exists).

Comment. I need a tiny bit of this formula, namely the terms with the components of $\nabla^{n-2}\mathrm{Rm}$ in the coefficients for the monomials of degree $n$.

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    $\begingroup$ Depending on how far you want to carry it out, for example, you could look at arxiv.org/pdf/0903.2087.pdf $\endgroup$ Dec 23, 2020 at 22:14
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    $\begingroup$ I can't help but wonder why you would want to use or refer to this formula. It's hard to imagine this arising in your work. $\endgroup$
    – Deane Yang
    Dec 23, 2020 at 22:21
  • $\begingroup$ Here's an initial comment: A monomial of degree $n$ has $n+2$ indices (2 more for the metric), and so does $\nabla^{n-2}\mathrm{Rm}$. So symmetry says that the terms in the coefficients of $x^{i_1}x^{i_2}\cdots x^{i_n}$ that have $\nabla^{n-2}\mathrm{Rm}$ in them have to be a constant times components of the projection of $\nabla^{n-2}\mathrm{Rm}$ onto the space of symmetric $(n+2)$-tensors. $\endgroup$
    – Deane Yang
    Dec 24, 2020 at 0:10
  • $\begingroup$ Do you need the constant? It's not obvious to me how to compute it. $\endgroup$
    – Deane Yang
    Dec 24, 2020 at 0:14
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    $\begingroup$ @DeaneYang I know how to find it: suppose that $I$ and $J$ are Jacobi fields with $0$ at $p$. Take expression for $\langle I,J\rangle^{(n+2)}$ at $p$. Set $t\cdot V= I$ and $t\cdot W= J$ and collect terms with $\langle(\nabla^{(n-2)}_T\mathrm{Rm})(V,T)T,W\rangle$ $\endgroup$ Dec 24, 2020 at 0:24

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Using the reference https://arxiv.org/pdf/0903.2087.pdf, which agrees with https://arxiv.org/pdf/hep-th/0001078v1.pdf, which agrees with the reference U. Müller, C. Schubert and Anton M. E. van de Ven, J. Gen. Rel. Grav. 31 (1999) 1759-1768 [arXiv], one sees that the expansion in normal coordinates about a point $p\in M$ of a metric $g$ is given by $$ g_{ij} = g_{ij}(0) + \sum_{n\ge 2} \frac1{n!} C_{ijk_1\cdots k_n}\,x^{k_1}\cdots x^{k_n} $$ where $C_{ijk_1\cdots k_n}$ is the value at $p$ of a tensor of the form $$ -\frac{2(n{-}1)}{n{+}1}\,\nabla_{k_3}\cdots\nabla_{k_n} R_{ik_1jk_2} + LOT_n $$ where $LOT_n$ is a polynomial in the curvature tensor and its derivatives of order strictly less than $n{-}2$. (Note that, in particular, $LOT_2 = 0$.)

Note: The first two references only verify the crucial coefficient $-2(n{-}1)/(n{+}1)$ as above for $n=2,3,4,5$, but, in any case, the coefficients for $n=2$ and $n=3$ are well-known.

Meanwhile, if one does not want to rely on these sources, one can note that these coefficients clearly cannot depend on the dimension of the underlying manifold, so it suffices to verify them in the simplest nontrivial case, i.e., when the dimension of the manifold is $2$. In this case, using the Gauss Lemma, a metric $g$ in geodesic normal coordinates $(x,y)$ centered on $p$ takes the form $$ g = \mathrm{d}x^2 + \mathrm{d}y^2 + h(x,y)\bigl(x\,\mathrm{d}y-y\,\mathrm{d}x)^2, $$ where the function $h$ is arbitrary, subject to the condition that $(x^2{+}y^2)h(x,y)+1>0$.

Letting $r^2 = x^2 + y^2$ and letting $T$ be the radial vector field $x\,\partial_x + y\,\partial_y$, one computes the formula for the Gauss curvature of $g$ to be $$ K = -\frac{2(1+r^2h)(TTh) - r^2(Th)^2+2(5+3r^2h)(Th) + 8r^2h^2+12h}{4(1+r^2h)^2}. $$ Of course, one has the formula for the Riemann curvature tensor in the form $$ R = K\,(1{+}r^2h)\,(\mathrm{d}x\wedge\mathrm{d}y)\otimes(\mathrm{d}x\wedge\mathrm{d}y), $$ while the tensor $(1{+}r^2h)\,(\mathrm{d}x\wedge\mathrm{d}y)\otimes(\mathrm{d}x\wedge\mathrm{d}y)$ is parallel with respect to the Levi-Civita connection. Then, by assuming that $h$ vanishes to order $n{-}2\ge0$, one can easily verify that the above coefficient is correct.

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  • $\begingroup$ This is cool. How do I peel out -2(n-1)/(n+1), let alone the form of the coefficient $C_*$, from the Gen.Rel.Grav. reference? Should we be able to read it off from their vielbein Equation 29? (Maybe I haven't put in enough effort.) $\endgroup$ Dec 24, 2020 at 21:24
  • $\begingroup$ An earlier reference: (2.34) in "Riemannian metrics with the prescribed curvature..." by Kowalski and Belger [Math. Nachr. 168 (1994)] $\endgroup$ Dec 25, 2020 at 1:59
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    $\begingroup$ @ChrisGerig: I'm not sure exactly what you are asking. Look at their definition of $\mathsf{R}_k$ in Equation (31) and then see how it is used in the formula for $g$ in the Appendix. Also, note their Equation (32). The minus sign in the formula I wrote above is because they use the first and last indices in the Riemann curvature tensor for the metric indices in the expansion while I use the first and third. Of course, $R_{ijkl}=-R_{ijlk}$. $\endgroup$ Dec 25, 2020 at 11:29
  • $\begingroup$ @AntonPetrunin: Thanks. I was limited to the references I could find and view online in a few minutes. It's not surprising that there are earlier references that give the explicit 'leading order' term in the Taylor expansion in normal coordinates. In principle, there's nothing hard about that (as you have pointed out), and it could have been written down any time in the last 100 years. Writing down the full expression (i.e., specifying the $LOT_n$), which is what I gathered you wanted (from the original version of your question), is more challenging. $\endgroup$ Dec 25, 2020 at 11:49
  • $\begingroup$ BTW, you say "strictly less than $n-2$" but it should be "less or equal". $\endgroup$ Dec 27, 2020 at 4:08

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