class_docstr: Class for (principal) fiber bundles.
  
  This class implements abstract methods for fiber bundles, or more
  generally manifolds, with a submersion map, or a right Lie group action.
  
  Parameters
  ----------
  group : LieGroup
      Group that acts on the total space by the right.
      Optional. Default : None.
      Either the group or the group action must be given.
  group_action : callable
      Right group action. It must take as input a point of the total space
      and an element of the group, and return a point of the total space.
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docstr: Compute the covariant derivative of the integrability tensor A.
  
  The covariant derivative :math:`\nabla_X (A_Y E)` in total space is
  necessary to compute the covariant derivative of the directional
  curvature in a submersion. The components :math:`\nabla_X (A_Y E)`
  and :math:`A_Y E` are computed at base-point :math:`P = base\_point`
  for horizontal vector fields :math:`X, Y` extending the values
  given in argument :math:`X|_P = horizontal\_vec\_x`,
  :math:`Y|_P = horizontal\_vec\_y` and a general vector field
  :math:`E` extending :math:`E|_x = tangent\_vec\_e`
  in a neighborhood of x with covariant derivatives
  :math:`\nabla_X Y |_P = nabla_x y` and
  :math:`\nabla_X E |_P = nabla_x e`.
  
  Parameters
  ----------
  horizontal_vec_x : array-like, shape=[..., {total_space.dim, [n, m]}]
      Horizontal tangent vector at `base_point`.
  horizontal_vec_y : array-like, shape=[..., {total_space.dim, [n, m]}]
      Horizontal tangent vector at `base_point`.
  nabla_x_y : array-like, shape=[..., {total_space.dim, [n, m]}]
      Tangent vector at `base_point`.
  tangent_vec_e : array-like, shape=[..., {total_space.dim, [n, m]}]
      Tangent vector at `base_point`.
  nabla_x_e : array-like, shape=[..., {total_space.dim, [n, m]}]
      Tangent vector at `base_point`.
  base_point : array-like, shape=[..., {total_space.dim, [n, m]}]
      Point of the total space.
  
  Returns
  -------
  nabla_x_a_y_e : array-like, shape=[..., {total_space.dim, [n, m]}]
      Tangent vector at `base_point`, result of :math:`\nabla_X
      (A_Y E)`.
  a_y_e : array-like, shape=[..., {ambient_dim, [n, n]}]
      Tangent vector at `base_point`, result of :math:`A_Y E`.
  
  References
  ----------
  .. [Pennec] Pennec, Xavier. Computing the curvature and its gradient
      in Kendall shape spaces. Unpublished.
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code: def integrability_tensor_derivative(
      self,
      horizontal_vec_x,
      horizontal_vec_y,
      nabla_x_y,
      tangent_vec_e,
      nabla_x_e,
      base_point,
  ):
      r"""Compute the covariant derivative of the integrability tensor A.
  
      The covariant derivative :math:`\nabla_X (A_Y E)` in total space is
      necessary to compute the covariant derivative of the directional
      curvature in a submersion. The components :math:`\nabla_X (A_Y E)`
      and :math:`A_Y E` are computed at base-point :math:`P = base\_point`
      for horizontal vector fields :math:`X, Y` extending the values
      given in argument :math:`X|_P = horizontal\_vec\_x`,
      :math:`Y|_P = horizontal\_vec\_y` and a general vector field
      :math:`E` extending :math:`E|_x = tangent\_vec\_e`
      in a neighborhood of x with covariant derivatives
      :math:`\nabla_X Y |_P = nabla_x y` and
      :math:`\nabla_X E |_P = nabla_x e`.
  
      Parameters
      ----------
      horizontal_vec_x : array-like, shape=[..., {total_space.dim, [n, m]}]
          Horizontal tangent vector at `base_point`.
      horizontal_vec_y : array-like, shape=[..., {total_space.dim, [n, m]}]
          Horizontal tangent vector at `base_point`.
      nabla_x_y : array-like, shape=[..., {total_space.dim, [n, m]}]
          Tangent vector at `base_point`.
      tangent_vec_e : array-like, shape=[..., {total_space.dim, [n, m]}]
          Tangent vector at `base_point`.
      nabla_x_e : array-like, shape=[..., {total_space.dim, [n, m]}]
          Tangent vector at `base_point`.
      base_point : array-like, shape=[..., {total_space.dim, [n, m]}]
          Point of the total space.
  
      Returns
      -------
      nabla_x_a_y_e : array-like, shape=[..., {total_space.dim, [n, m]}]
          Tangent vector at `base_point`, result of :math:`\nabla_X
          (A_Y E)`.
      a_y_e : array-like, shape=[..., {ambient_dim, [n, n]}]
          Tangent vector at `base_point`, result of :math:`A_Y E`.
  
      References
      ----------
      .. [Pennec] Pennec, Xavier. Computing the curvature and its gradient
          in Kendall shape spaces. Unpublished.
      """
      raise NotImplementedError
  
file_path: geomstats/geomstats/geometry/fiber_bundle.py
imports: import sys
  from abc import ABC
  from scipy.optimize import minimize
  import geomstats.backend
  from geomstats.vectorization import get_batch_shape
label_desc: Use image node_id 12 for calling the FiberBundle obj's underlying member method code with example usage: obj.integrability_tensor_derivative(horizontal_vec_x, horizontal_vec_y, nabla_x_y, tangent_vec_e, nabla_x_e, base_point) without return types