I'm working on a numerical optimization problem that naturally lives on the Grassmann Manifold Gr$_N(\mathbb{C^M})$, however the objective function is defined on the alternating algebra given by the Plucker embedding $\Omega: $Gr$_N(\mathbb{C^M}) \rightarrow \mathbb{P}(\wedge^N \mathbb{C}^N)$.

A lot work has been done on optimization on Grassmann manifolds, and while there is some choice here, a typical starting point is to define a metric in the tangent space of a point $X$ as the restriction of the Euclidean inner product to the tangent space $T_X$. As such, for $V, W \in T_X($Gr$_N(\mathbb{C^M}))$ we have $\langle V, W \rangle = $ Tr $V^H W$. The alternating algebra, on the other hand, has a standard inner product such that for $v=\wedge_i^N v_i, w=\wedge_i^N w_i \in \mathbb{P}(\wedge^N \mathbb{C}^N)$ we have $\langle v,w \rangle = \det( \langle v_i, w_j \rangle )$.

This seems to suggest an alternative possible metric on the Grassmann manifold, given by taking $\langle V, W \rangle = \langle \Omega(V), \Omega(W) \rangle = \det(V^H W)$. Is this a well-defined metric that has been studied before?

I'm particularly interested because the objective function also lives in this space, and being an algebra instead of a manifold, there seems to be (at least from a superficial inspection) less issues with finding relations between vectors in the tangent spaces of different points. That is, parallel transport seems to be less of an issue. If anyone is aware of resources exploring this topic, preferably an applied treatment with some relation to optimization, I would be greatly appreciative, however any information is welcome. If this is a wildly misguided direction, I would also appreciate insight on that front.

Thank you.


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