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My question is the following :

Given an algebraic curvature operator $R\in S^2_B(\Lambda^2\mathbb{R}^n)$, is there an a simple criterion to know if this curvature operator can occur as the curvature operator of symmetric space ?

I would be almost equally happy if someone can point to a way to know if $R$ can be the curvature operator of some Riemannian manifold with reduced holonomy $G\subset SO(n,\mathbb{R})$.

I suspect this might be linked to the fact that $\Lambda^2\mathbb{R}^n=\mathfrak{so}(n,\mathbb{R})$ and how $R$ acts on $\mathfrak{g}\subset\mathfrak{so}(n,\mathbb{R})$. So far, the only link I found is that the image of $R$ has to be contained in $\mathfrak{g}$. I have no idea if this is sufficient. And I wonder if it is possible to write a condition NOT along the line of "There is a Lie subalgebra $\mathfrak{g}\subset\mathfrak{so}(n,\mathbb{R})$ such that..."

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What is $S_B^2$? –  plusepsilon.de Jan 25 '13 at 14:17
    
I was about to edit my question to add more details, but Pr Bryant just says what it means at the beginning of his answer ! –  Thomas Richard Jan 25 '13 at 15:14

1 Answer 1

up vote 22 down vote accepted

I suppose that $S^2_B(\Lambda^2(\mathbb{R}^n))$ means the subspace of $S^2(\Lambda^2(\mathbb{R}^n))$ that satisfies the Bianchi identity, i.e., the kernel of the natural map $S^2(\Lambda^2(\mathbb{R}^n))\longrightarrow \Lambda^4(\mathbb{R}^n)$.

Certainly, you'd need that, if ${\frak{g}}\subset \Lambda^2(\mathbb{R}^n)$ is the smallest subspace such that $R$ lies in $S^2({\frak{g}})$ (and this ${\frak{g}}$ is unique and easily computable from $R$), then ${\frak{g}}$ has to be the Lie algebra of a closed, connected subgroup $G\subset \mathrm{SO}(n)$.

This is not enough, though, as the case $n=3$ shows: The generic $R$ in this case has ${\frak{g}}=\Lambda^2(\mathbb{R}^3)={\frak{so}}(3)$ but isn't the curvature of a symmetric space.

In addition, one needs that $R$ be invariant under the action of this $G$ on $S^2(\Lambda^2(\mathbb{R}^n))$. Conversely, if this invariance holds, then $R$ is the curvature of an irreducible $n$-dimensional symmetric space with holonomy $G$.

The reason is that one then can define a Lie algebra structure on ${\frak{l}} = {\frak{g}}\oplus \mathbb{R}^n$ as follows: Let the bracket on ${\frak{g}}\subset{\frak{l}}$ be the usual bracket on ${\frak{g}}$; for $a\in{\frak{g}}\subset{\frak{so}}(n)$ and $x\in \mathbb{R}^n$, set $[a,x]=-[x,a]=ax$; and for $x$ and $y$ in $\mathbb{R}^n$, set $[x,y]= R(x{\wedge}y)$. (Here, we are regarding $R$ as a symmetric mapping $R:\Lambda^2(\mathbb{R}^n)\to \Lambda^2(\mathbb{R}^n)$, knowing that it has image in ${\frak{g}}\subset \Lambda^2(\mathbb{R}^n)$.) Then the assumption that $R$ lies in $S^2_B(\Lambda^2(\mathbb{R}^n))$ is just that $R(x{\wedge}y)z+R(y{\wedge}z)x+R(z{\wedge}x)y=0$ while the assumption that $R$ is invariant under $G$ implies that $a\ R(x{\wedge}y)-R(x{\wedge}y)\ a = R(ax{\wedge}y + x{\wedge}ay)$, and these are exactly the equations needed to verify that the bracket defined as above on ${\frak{l}}$ satisfies the Jacobi identity. Then the pair $({\frak{l}},{\frak{g}})$ is a symmetric pair, so that, by Cartan's construction, there is an $n$-dimensional Riemannian symmetric space $M=L/G$ with holonomy $G$ and having $R$ as its curvature operator.

Added remark: The OP wanted a criterion that didn't explicitly mention a Lie algebra $\frak{g}$, and one can do it this way: If one just defines $\frak{g}$ to be the image of $R$ in $\Lambda^2(\mathbb{R}^n)$ when one regards $R$ as a symmetric map from $\Lambda^2(\mathbb{R}^n)$ to itself, then the condition that $$ R(w{\wedge}z)\ R(x{\wedge}y)-R(x{\wedge}y)\ R(w{\wedge}z) = R\bigl(R(w{\wedge}z)x{\wedge}y\bigr) + R\bigl(x{\wedge}R(w{\wedge}z)y\bigr) $$ hold for all $x,y,w,z\in\mathbb{R}^n$ implies both that $\frak{g}$ be closed under Lie bracket (so that $\frak{g}$ is a Lie algebra and that $R$ be invariant under the action of the connected Lie subgroup $G\subset\mathrm{SO}(n)$ whose Lie algebra is $\frak{g}$. Thus, the above system of (quadratic) equations on $R$ is exactly the algebraic condition that $R\in S^2_B\bigl(\Lambda^2(\mathbb{R}^n)\bigr)$ be the curvature operator of a symmetric space. It's also, not surprisingly, the condition that the bracket defined above on $\frak{l}=\frak{g}\oplus\mathbb{R}^n$ satisfy the Jacobi identity. This may be more along the lines of what the OP had in mind with his question.

Your other question about characterizing curvature operators of Riemannian manifolds with reduced holonomy is not as easy to answer. You can, of course, tell when $R$ lies in $S^2({\frak{g}})\subset S^2(\Lambda^2(\mathbb{R}^n))$ for a given Lie algebra ${\frak{g}}\subset{\frak{so}}(n)=\Lambda^2(\mathbb{R}^n)$, but then knowing whether there is a metric with holonomy $G$ whose curvature tensor takes the value $R$ at some point is a little tricky.

In the special case that $\frak{g}$ acts irreducibly on $\mathbb{R}^n$ and can be the holonomy of a Riemannian metric that is not locally symmetric, there is a result (I guess, due to me in several cases) that asserts that, in fact, such an $R$ does occur as the curvature operator at some point of a Riemannian metric with holonomy $G\subset\mathrm{SO}(n)$. I proved this using Cartan-Kähler theory for the exceptional cases $\mathrm{G}_2\subset\mathrm{SO}(7)$ and $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$ in Metrics with exceptional holonomy, Ann. of Math. (2) 126 (1987). It is obvious in the cases of $\mathrm{SO}(n)$ itself and $\mathrm{U}(n)\subset\mathrm{SO}(2n)$ and easy in the case of $\mathrm{SU}(n)\subset\mathrm{SO}(2n)$. I gave proofs (not in all detail, I admit) for the remaining cases $\mathrm{Sp}(n)$ and $\mathrm{Sp}(n)\mathrm{Sp}(1)$ in $\mathrm{SO}(4n)$ in Classical, exceptional, and exotic holonomies: a status report, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 93–165, Sémin. Congr., 1, Soc. Math. France, Paris, 1996, again using Cartan-Kähler theory.

However, it seems unlikely (though I don't know an explicit counterexample off the top of my head) that you could find a metric such that the curvature operator at every point is equivalent to $R$ up to $G$-conjugacy. (Probably, this is not possible even for $\mathrm{SU}(2)\subset\mathrm{SO}(4)$, but I'd have to think about it to be sure.)

Added remark: I have now checked, and, indeed, there is no $4$-dimensional Riemannian manifold with holonomy $\mathrm{SU}(2)$ such that the curvature operators $R_p$ at all points $p\in M$ are conjugate under $\mathrm{SO(4)}$. [Note that this condition is a priori weaker than the condition of being homogeneous; rather it's just the assumption that the germs of the metric at all points agree up to second order, i.e., that the metric is what is sometimes called 'curvature homogeneous'. (By comparison, there are many Riemannian $3$-manifolds such that all of the eigenvalues of the curvature are constant and so are 'curvature homogeneous', but that have no nontrivial Killing fields.)]

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Thanks for the explanation of S2B:-) –  plusepsilon.de Jan 25 '13 at 15:15
    
Ok, thank you for this detailed answer. In the first case, how would one construct the symmetric space which has $R$ as curvature operator ? Would you have a reference for these kind of results ? Thanks again. –  Thomas Richard Jan 25 '13 at 15:19
    
Sure. I'll put the first construction in the answer. The existence result that I allude to in the last paragraph is a bit scattered in a few different papers, but I'll try to put in some references for that as well. –  Robert Bryant Jan 25 '13 at 16:04
    
Thanks again ! That's clear now. –  Thomas Richard Jan 25 '13 at 17:19
    
Even better. Thanks. –  Thomas Richard Jan 27 '13 at 10:41

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