Does a connection satisfying $(\nabla_X g)(Y,Z)-(\nabla_Y g)(X,Z)=0$ have a special name or satisfy a special property?

Question

Let $\nabla$ be a flat torsion-free connection on a smooth manifold. Let $g$ be a metric on $M$ ($\nabla$ may not be the Levi-Civita connection of $g$). Suppose that

$$ (\nabla_X g)(Y,Z)-(\nabla_Y g)(X,Z)=0. $$ Does $\nabla$ receive a special name? Or does it imply $\nabla$ is related with the Levi-Civita connection or any other thing stronger than the above relation itself? Maybe the existence of some coordinates, I don't know.

I ask this because in Dombrowski's paper On the geometry of the Tangent Bundle, it is proved that the presymplectic form defined as

$$\omega(X,Y)=g_S(X,JY),$$

where $J$ is the canonical complex form on $TM$ defined using the split induceb by $\nabla$ and $g_S$ denotes the Sasaki metric defined in terms of the same split and the metric $g$, is closed whenever $\nabla$ is the Levi-Civita connection.

However, the computation for the general case shows that the $d\omega$ is proportional to the above factors when evaluated on some vector fields. But I'm sure the $\omega$ is closed. In fact, I would say, under certain conditions, it is the pullback of the canonical 2-form on $T^*M$ along $g$. But I don't know exactly what conditions are necessary and if they are related to my expression above.

P.S. It would be useful also to recieve some suggestions of references to be able to read about this.

Addendum 1.

I typed the question on the phone. Let me describe in detail Dombrowski's construction now I have a computer avaible.

It is well known every linear connection on $M$ defines an split on $TTM$ into vertical and horizontal subbundles. Each of them is isomorphic to $TM$ and for every vector field $Z\in\Gamma(TTM)$ there are vector fields $X,Y\in\Gamma(TM)$ such that $Z=X^h+Y^v$, where the superscripts v and h denote the verticla and the horizontal litf, respectively. Hence, every tensor on $TM$ is completely characterised once we know how it acts on horizontal and vertical vector fields.

Now, Dombroski defines a complex tructure $J:TTM\rightarrow TTM$ as

$$JX^h, X^v, \qquad JX^v= - X^h. $$

He proves $J$ is integrable if and only if $\nabla$ is flat and torsion-free. But in the third appendix we goes beyon. The aim is to prove $TM$ is a Kähler. For that he considers a Riemannian metric $g$ on $M$ (again the flat and torsion-free connection $\nabla$ may not be the Levi-civita connection for $g$) and define the Sasaki metric

$$ g_S(X^h,Y^h)=g_S(X^v,Y^v)= \pi_{TM}\circ g(X,Y), \qquad g_S(X^h,Y^v)=g_S(X^v,Y^h)=0 $$

(Notice I compose on the right and I denote the differential of a map $f$ as $Tf$). It is not difficult to see that $J$ leaves $g_S$ invariant. Hence $\omega(X,Y)=g_S(X,JY)$ is 2-form.

Finally, he particularise for the case when $\nabla$ is the Levi-Civita connection for $g$. In this case he proves $\omega$ is locally and non-locally an exact 2-form. But the computation can be done for a non-metric connection. In that case one finds the relation I have written above.

Addendum 2.

I claim under certain conditions, $\omega$ is the pullback along $g$ of the canonical 2-form defined on $T^*M$. For that, let me write firstly the pullback of the canonical 1-form:

$$ g^*\theta(Z)|_{p,X}= g_p(X,(T\pi)_{(p,X)}Z), \qquad Z\in T_{(p,X)}TM. $$

In coordinates $g^*\theta$ reads simply as $y_ig_{ij}d\tilde{x}_j$. If I compute the diferential I get

$$ d(g^*\theta)= -g_{ij}d\tilde{x}_i\wedge dy_j -y_i\frac{\partial g_{ij}}{\partial x_k} d\tilde{x}_i\wedge d\tilde{x}_k, $$ while $\omega$ reads in coordinates as $-g_{ij}d\tilde{x}_i\wedge dy_j$ (or the opposite, I'm not 100% sure about the sign). Since the connection has not been used, it is logic to expect $dg^*\theta$ cannot match always with the above $\omega$. However, there are certain conditions under it should. What are they? Is it related to the relations I am asking for?

Answer 1

I'm not familiar with the specific construction you're studying, I don't have a direct answer, but this will be too long for a comment. If $E \to M$ is a vector bundle with connection $\nabla$, you can define a exterior derivative ${\rm d}^\nabla\colon \Omega^k(M;E) \to \Omega^{k+1}(M;E)$ acting on $E$-valued differential forms, with the aid of $\nabla$. Namely, in the usual formula for the exterior derivative, everytime a field $X_i$ is acting on a smooth function, you write $\nabla_{X_i}$ instead. So, e.g., $${\rm d}^\nabla\alpha(X,Y) = \nabla_X\alpha(Y) - \nabla_Y\alpha(X) - \alpha([X,Y]),$$and so on. Note that we do not have ${\rm d}^\nabla \circ {\rm d}^\nabla = 0$ is $\nabla$ is not flat. Any $B \in \Gamma(T^*M\otimes E^*)$ (that is, is $B\colon \mathfrak{X}(M)\times \Gamma(E) \to \mathscr{C}^\infty(M)$ is $\mathscr{C}^\infty(M)$-bilinear) can be seen as a $E^*$-valued $1$-form, by $$X \mapsto B(X,\cdot),$$and so it makes sense to talk about the exterior derivative of that. We have $$({\rm d}^\nabla B)(X,Y)\psi = (\nabla_XB)(Y,\psi) - (\nabla_YB)(X,\psi),$$whenever we use any auxiliary torsion-free connection in $TM$ to form covariant derivatives of $B$. When $E = TM$, we say that $B$ is a Codazzi tensor if ${\rm d}^\nabla B = 0$. So in your case, you're looking for the connections which make the metric itself a Codazzi tensor.