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Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is the analytic subgroup of $G$ with Lie algebra $\mathfrak{k}$ in a Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$.

I have computed an explicit formula for the Levi-Civita connection on $T(G/K)$ in terms of the Lie-algebraic structure $\mathfrak{g}$, and searched quite a long time for a reference to check the formula against, without success.

What I mean is the following. Write $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{a}\oplus \mathfrak{n}$, $G=KAN$ by the Iwasawa decomposition. For $X\in \mathfrak{g}$ and $g\in G$, write $$ k_I(X)=q,\qquad K_I(g)=k $$ if $X=q+r+s$ with $q\in \mathfrak{k}$, $r\in \mathfrak{a}$, $s\in \mathfrak{n}$ and $g=kan$ with $k\in K$, $a\in A$, $n\in N$.

A vector field $\mathcal{X}$ on $G/K$ can be identified with a right-$K$-equivariant function $\bar{\mathcal{X}}:G\to \mathfrak{p}$, where $K$ acts on $\mathfrak{p}$ by the adjoint representation. It is easy to see that for any Lie algebra element $X\in \mathfrak{p}$, we obtain an associated vector field $\mathcal{X}(X)$ by setting $$ \bar{\mathcal{X}}(X)(g):=\mathrm{Ad}(K_I(g^{-1}))\,X,\qquad g\in G. $$ Let now $\{X_i\}$ be an orthonormal basis of $\mathfrak{p}$ with respect to the Killing form. Then the associated vector fields $\mathcal{X}(X_i)=:\mathcal{X}_i$ span a global orthonormal frame of the tangent bundle $T(G/K)$. To know the Levi-Civita connection explicitly, it suffices to know the covariant derivatives $\nabla_{\mathcal{X}_i}\mathcal{X}_j$.

Computing the latter directly using the Koszul formula, I find after tedious calculations (there might be a sign error) $$ \nabla_{\mathcal{X}_i}\mathcal{X}_j=\mathcal{X}(X_{i,j}),\qquad X_{i,j}=[X_j,k_I(X_i)]. $$ To my astonishment I cannot find a similar formula in the standard books on the topic by Helgason or Knapp, or anywhere else using google.

I am sure that an expert in the field knows where to find an explicit formula for the covariant derivatives $\nabla_{\mathcal{X}_i}\mathcal{X}_j$, perhaps in a different formulation (which I hope would be equivalent to my result above). There might also be a very easy way to deduce the required formula which I didn't see.

What I find remarkable about the formula above is that it shows:

The geometry of the Riemannian symmetric space $G/K$ is a manifestation of the non-compatibility of the Cartan and Iwasawa decompositions of the Lie algebra $\mathfrak{g}$ of $G$.

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  • $\begingroup$ What does the index $I$ in $K_I$ and $\mathfrak{k}_I$ mean ? If $X_i$ is a basis of $\mathfrak{p}$, then shouldn't be $k_I(X_i) = 0$? $\endgroup$ – Vít Tuček Oct 25 '17 at 7:26
  • $\begingroup$ The $I$ is just supposed to mean "Iwasawa". The point is that $k_I$ is not an orthogonal projection onto $\mathfrak{k}$ because the Iwasawa decomposition is not orthogonal. Therefore, an element $X\in\mathfrak{p}$ can have $k_I(X)\neq 0$ even though it is contained in the orthogonal complement of $\mathfrak{k}$. $\endgroup$ – B K Oct 25 '17 at 8:31

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