class_docstr: Procrustes metric on the pre-shape space.
class_docstr_tok: ["Procrustes","metric","on","the","pre-shape","space","."]
docstr: Compute the covariant derivative of the curvature.
  
  For four vectors fields :math:`H|_P =` `tangent_vec_a`,
  :math:`X|_P =` `tangent_vec_b`, :math:`Y|_P =` `tangent_vec_c`,
  :math:`Z|_P =` `tangent_vec_d` with
  tangent vector value specified in argument at the `base_point` :math:`P`,
  the covariant derivative of the curvature
  :math:`(\nabla_H R)(X, Y) Z |_P` is computed at the `base_point` :math:`P`.
  Since the sphere is a constant curvature space this vanishes identically.
  
  Parameters
  ----------
  tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
      Tangent vector at `base_point` along which the curvature is
      derived.
  tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
      Unused tangent vector at `base_point` (since curvature derivative
      vanishes).
  tangent_vec_c : array-like, shape=[..., k_landmarks, m_ambient]
      Unused tangent vector at `base_point` (since curvature derivative
      vanishes).
  tangent_vec_d : array-like, shape=[..., k_landmarks, m_ambient]
      Unused tangent vector at `base_point` (since curvature derivative
      vanishes).
  base_point : array-like, shape=[..., k_landmarks, m_ambient]
      Unused point on the group.
  
  Returns
  -------
  curvature_derivative : array-like, shape=[..., k_landmarks, m_ambient]
      Tangent vector at `base_point`.
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code: def curvature_derivative(
      self,
      tangent_vec_a,
      tangent_vec_b=None,
      tangent_vec_c=None,
      tangent_vec_d=None,
      base_point=None,
  ):
      r"""Compute the covariant derivative of the curvature.
  
      For four vectors fields :math:`H|_P =` `tangent_vec_a`,
      :math:`X|_P =` `tangent_vec_b`, :math:`Y|_P =` `tangent_vec_c`,
      :math:`Z|_P =` `tangent_vec_d` with
      tangent vector value specified in argument at the `base_point` :math:`P`,
      the covariant derivative of the curvature
      :math:`(\nabla_H R)(X, Y) Z |_P` is computed at the `base_point` :math:`P`.
      Since the sphere is a constant curvature space this vanishes identically.
  
      Parameters
      ----------
      tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
          Tangent vector at `base_point` along which the curvature is
          derived.
      tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
          Unused tangent vector at `base_point` (since curvature derivative
          vanishes).
      tangent_vec_c : array-like, shape=[..., k_landmarks, m_ambient]
          Unused tangent vector at `base_point` (since curvature derivative
          vanishes).
      tangent_vec_d : array-like, shape=[..., k_landmarks, m_ambient]
          Unused tangent vector at `base_point` (since curvature derivative
          vanishes).
      base_point : array-like, shape=[..., k_landmarks, m_ambient]
          Unused point on the group.
  
      Returns
      -------
      curvature_derivative : array-like, shape=[..., k_landmarks, m_ambient]
          Tangent vector at `base_point`.
      """
      batch_shape = get_batch_shape(
          self._space.point_ndim,
          tangent_vec_a,
          tangent_vec_b,
          tangent_vec_c,
          tangent_vec_d,
          base_point,
      )
      return gs.zeros(batch_shape + self._space.shape)
  
imports: import geomstats.backend
  from geomstats.geometry.base import LevelSet
  from geomstats.geometry.fiber_bundle import FiberBundle
  from geomstats.geometry.hypersphere import Hypersphere
  from geomstats.geometry.matrices import Matrices
  from geomstats.geometry.quotient_metric import QuotientMetric
  from geomstats.geometry.riemannian_metric import RiemannianMetric
  from geomstats.integrator import integrate
  from geomstats.vectorization import get_batch_shape, repeat_out
label_desc: Use image node_id 6 for calling the PreShapeMetric obj's underlying member method code with example usage: obj.curvature_derivative(tangent_vec_a, tangent_vec_b, tangent_vec_c, tangent_vec_d, base_point) and returns: gs