Kulkarni-Nomizu square root of the Riemann tensor

Question

Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of Proposition 15 (Chapter 4, Section 2, page 99) in the book of Petersen "Riemannian geometry" - Second Edition, that positivity of the curvature operator ${\mathscr{R}}$ on $\Lambda^2T_pM$ is sufficient in dimension 3. Moreover, when $n=3$, $\det{\mathscr{R}}>0$ is sufficient.
Anyone have a reference for $n>3$, please?

This is how to know if there exists a square matrix $n\times n$ with assigned all its $2\times2$ subdeterminants (and possibly how to recover it). In the following paper of Greenhill http://web.maths.unsw.edu.au/~csg/papers/ext-matrix.pdf, an algorithm to find such a matrix is showed, but apparently it does not state a theoretical ("easy") criterium for its existence.

My question is related to my other question 2x2 subdeterminants of a matrix which was related to uniqueness.

Answer 1

Note that Petersen's Prop 15 does not directly address your question. The assumption that there already exists an embedding from the $n$ dimensional $M$ to $(n+1)$ dimensional Euclidean space is crucial. Petersen's Proposition 15 concerns not solvability of $\mathrm{Riem} = S\star S$ but whether the solution $S$ is unique.

What you are looking for is a theorem due to Jaak Vilms. He described it in two different places:

1. Local isometric imbedding of Riemannian $n$-manifolds into Euclidean $(n+1)$-space, J. Differential Geom. 12(2): 197-202 (1977). DOI: 10.4310/jdg/1214433981
2. Factorization of Curvature Operators, Transactions of the American Mathematical Society Vol. 260, No. 2 (Aug., 1980), pp. 595-605. https://www.jstor.org/stable/1998025

Note that this discussion presumes that you have a given Riemannian metric (or I guess in your case a positive definite inner product on $T_pM$).

Essentially: if one looks at the Gauss-Codazzi equations, one sees that when $M$ has an isometric immersion into Euclidean space of $p$ dimensions higher, then one has a decomposition of $$ \mathrm{Riem} = \sum_{i = 1}^p S_i \star S_i $$ And so the question you are asking for is identical the (local) isometric immersivity of $M^n$ into $E^{n+1}$.

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Notation

Denote by $\mathscr{R}: \Lambda^2 T_pM \to \Lambda^2 T_pM$ the curvature operator acting on two forms. In indices its components are $R^{ij}_{kl}$ antisymmetric in $i,j$ and in $k,l$ separately.

Define $$ \phi(\mathscr{R}) = R^{ij}_{kl} R^{kp}_{iq} R^{lq}_{jp} $$ and $$ \psi(\mathscr{R}) = R^{ij}_{kl} R^{kl}_{pq} R^{pq}_{ij} $$

The curvature operator $\mathscr{R}$ is assumed to satisfy Bianchi identity.

The curvature operator $\mathscr{R}$ is said to "preserve decomposability" if for arbitrary 2 forms $\alpha, \beta$: $$ \mathscr{R}\alpha \wedge \mathscr{R}\beta = 0 \iff \alpha \wedge \beta = 0 $$ Note that this condition is trivial if $n = 3$.

Given a curvature operator $\mathscr{R}$, its rank subspace is the smallest subspace $W$ of $T_pM$ such that the image of $\mathscr{R}$ lies in $\Lambda^2 W$; below when we refer to rank of $\mathscr{R}$ we mean the dimension of $W$.

Theorem

A curvature operator with rank $> 1$ decomposes as $S\star S$ if and only if

1. $\mathcal{R}$ preserves decomposability
2. $\phi + \frac14 \psi > 0$; in the case $n = 3 \pmod 4$ this condition can be replaced by positivity of $\det \mathscr{R}$ when restricted to the complement of its kernel.
3. a technical "non singularity condition".

The non-singularity condition is a bit annoying to state; see the second paper for details. It is trivially satisfied when $\mathscr{R}$ is positive (or negative) definite, and $n$ is either 3 or $>4$. (I think it may also be satisfied when $n = 4$ with definite curvature operator, but the computations are less obvious.)