cyjax.induced_metric

cyjax.induced_metric(metric, grad_def, dependent, grad_def_c=None)

Induced metric in local patch coordinates of variety.

If \(x\) are local coordinates and the \(z\) corresponding affine coordinates in ambient space, this function computes the pullback of a metric \(g_{ij}\) to the variety given the gradients \(\partial P / \partial z_i\) of its defining equation:

\[g^X_{k\bar{l}} = \frac{\partial z^{(p)}_{i}}{\partial z^{(p)}_{k}} \frac{\partial \bar{z}^{(p)}_{\bar{\jmath}}} {\partial \bar{z}^{(p)}_{\bar{l}}} g_{i\bar{\jmath}}\]

With the ambient projective space being \(d+1\) dimensional, the metric \(g_{ij}\) in affine coordinates has indices \(i, j\) ranging from 0 to \(d\). Local coordinates on the variety are defined by omitting index local_dep_index from the affine coordinate vector. The pullback of the metric will thus have one fewer value in each index \(k, l\).

Parameters:

• metric (Union[Array, ndarray, bool_, number]) – Two-dimensional \((d+1) \times (d+1)\) array specifying the metric in ambient affine coordinates.

• dependent (int) – Index of the dependent coordinate in the affine coordinate vector.

• grad_def (Union[Array, ndarray, bool_, number]) – Array \(dP / dz_i\) where \(P(z) = 0\) is the defining polynomial.

• grad_def_c (Union[Array, ndarray, bool_, number, None]) – Optionally pass the complex conjugate of the gradient of the defining equation to avoid re-computation.

Return type:

Union[Array, ndarray, bool_, number]

Returns:

A \(d \times d\) matrix; pullback of the ambient metric to local coordinates.

See Also: jacobian_embed()