Understand Riemannian cross-derivative on product manifolds

Suppose we have a smooth function $$f:\mathcal{M}\times\mathcal{N}\rightarrow\mathbb{R}$$ where the domain is a product of two Riemannian manifolds. The Riemannian cross-derivative ([1], section 2) is defined as: $$\mathrm{grad}_{xy}^2:= \mathrm{D}_x\mathrm{grad}_y f(x,y): \mathrm{T}_x\mathcal{M}\rightarrow \mathrm{T}_y\mathcal{N}$$

I'm trying to understand this operator. I have two specific questions:

First, to me, if we fix $$y$$ then $$\mathrm{grad}_y f(x,y):\mathcal{M}\rightarrow\mathrm{T}\mathcal{N}$$, and the differential on $$x$$ would result in an operator $$\mathrm{D}_x\mathrm{grad}_y f(x,y):\mathrm{T}\mathcal{M}\rightarrow\mathrm{T}\mathrm{T}\mathcal{N}$$, which seems not the same as the definition in the paper. Is there any point I missed here?

Second, I'm trying to understand it similar to its Euclidean counterparts. Under what curcumstances do we know that $$\mathrm{grad}_{xy}^2$$ and $$\mathrm{grad}_{yx}^2$$ are adjoints? That's to say, do we have: $$\langle \eta,\mathrm{grad}_{xy}^2(\xi) \rangle_{y} = \langle\mathrm{grad}_{yx}^2(\eta),\xi \rangle_{x}, \forall \xi\in \mathrm{T}_{x}\mathcal{M}\text{ and }\forall \eta\in \mathrm{T}_{y}\mathcal{N}$$

Any comments or references are appreciated, thanks!

References:

[1] Han, Andi, et al. "Riemannian Hamiltonian methods for min-max optimization on manifolds." SIAM Journal on Optimization 33.3 (2023): 1797-1827.

If you fix $$y$$, the function $$x \mapsto \mathrm{grad}_y f(x,y)$$ is a function mapping $$M \to T_y N$$; note that the codomain is a single, fixed vector space. So the differential in $$x$$ is a mapping $$TM \to T(T_yN)$$, but as $$T_yN$$ is a linear space its tangent space is canonically isomorphic to itself.
They should always be adjoints. You have, extending $$\eta$$ to a constant map $$M \to T_yN$$, $$\langle \eta, \mathrm{grad}_{xy}^2(\xi)\rangle_y = \xi( \langle \eta, \mathrm{grad}_y f(x,y) \rangle_y) = \xi(\eta(f))$$ So: given a vector $$\eta\in T_yN$$ and a vector $$\xi$$ in $$T_xM$$, extend them to a vector field $$\eta$$ on $$N$$ and a vector field $$\xi$$ on $$M$$, and then extend them trivially to vector fields on $$\tilde{\eta}(x,y) = (0,\eta(y))$$ and $$\tilde{\xi}(x,y) = (\xi(x),0)$$ on $$M\times N$$. Then our computation above shows that the difference between the two expression you are asking about is exactly $$[\tilde{\xi},\tilde{\eta}]f$$, but it is easy to check that this Lie Bracket vanishes.
• Another way to think about items 1 and 2: your manifold inherits a product Riemannian structure and hence a Levi-Civita connection, with respect to which you can define the Hessian of $f$. The musical operations allow you to consider the operation $H^\sharp_f: T(M\times N) \to T(M\times N)$, which is self-adjoint since the Levi-Civita connection is torsion free. The product structure gives the canonical decomposition $T_{x,y}(M\times N) = T_xM \times T_y N$ and your cross derivative is just the off-diagonal parts of $H^\sharp_f$. Commented Sep 26, 2023 at 6:07