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I saw the following question somewhere on the forum—I don't remember which thread exactly—and I think it even went unanswered.
Consider a manifold ##M## with local coordinates ##t=(t^1,\ldots,t^m)##, equipped with a symmetric connection ##\Gamma_{ij}^k(t)##. What sufficient conditions must be imposed on a connection for it to be generated by a Riemannian metric?
And in this regard, I would like to recall the following theorem.
Let ##T\subset\mathbb{R}^m,\quad D\subset \mathbb{R}^p## be open domains and let ##f_i^j:T\times D\to \mathbb{R}## be smooth functions. Consider an initial value problem
$$\frac{\partial z^j}{\partial t^i}=f_i^j(t,z),\quad z(\hat t)=\hat z,\quad (\hat t,\hat z)\in T\times D,\quad z=(z^1,\ldots,z^p).$$
Theorem. This problem has a unique solution ##z=z(t)## (locally in ##t##) provided the following equations hold:
$$\frac{df^j_i}{dt^k}=\frac{df^j_k}{dt^i},\quad \frac{df^j_i}{dt^k}:=\frac{\partial f^j_i}{\partial t^k}+\frac{\partial f^j_i}{\partial z^r}f^r_k.$$
This theorem should be applied to the equation ##\nabla_ig_{sj}(t)=0## with ##z=\{g_{ij}\}##.
Here, of course, the problem of passing from the local formulation to the global one remains.
Consider a manifold ##M## with local coordinates ##t=(t^1,\ldots,t^m)##, equipped with a symmetric connection ##\Gamma_{ij}^k(t)##. What sufficient conditions must be imposed on a connection for it to be generated by a Riemannian metric?
And in this regard, I would like to recall the following theorem.
Let ##T\subset\mathbb{R}^m,\quad D\subset \mathbb{R}^p## be open domains and let ##f_i^j:T\times D\to \mathbb{R}## be smooth functions. Consider an initial value problem
$$\frac{\partial z^j}{\partial t^i}=f_i^j(t,z),\quad z(\hat t)=\hat z,\quad (\hat t,\hat z)\in T\times D,\quad z=(z^1,\ldots,z^p).$$
Theorem. This problem has a unique solution ##z=z(t)## (locally in ##t##) provided the following equations hold:
$$\frac{df^j_i}{dt^k}=\frac{df^j_k}{dt^i},\quad \frac{df^j_i}{dt^k}:=\frac{\partial f^j_i}{\partial t^k}+\frac{\partial f^j_i}{\partial z^r}f^r_k.$$
This theorem should be applied to the equation ##\nabla_ig_{sj}(t)=0## with ##z=\{g_{ij}\}##.
Here, of course, the problem of passing from the local formulation to the global one remains.
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