Hadamard product

[Petrichor]

Hello,

While reading this article https://ui.adsabs.harvard.edu/abs/2022SigPr.19008322H/abstract

I stumbled open the equation (15). The thing is that I cannot see why there is a Hadamard product! (why does it appear suddenly). The metric should be just the inner space, am I wrong?

Thank you.

[Nicolas Boumal]

Equation 15 describes the tangent space, not the metric.

The conditions for a vector to be in a tangent space are really one condition that needs to be satisfied by each component of that vector.

The Hadamard product is just a convenient way to express these individual conditions all at once.