code: def integrability_tensor_alt(
      bundle, tangent_vec_a, tangent_vec_b, base_point
  ):
      r"""Compute the fundamental tensor A of the submersion.
  
      The fundamental tensor A is defined for tangent vectors of the total
      space by [ONeill]_ :math:`A_X Y = ver\nabla^M_{hor X} (hor Y)
      + hor \nabla^M_{hor X}( ver Y)` where :math:`hor,ver` are the
      horizontal and vertical projections.
  
      For the pre-shape space, we have closed-form expressions and the result
      does not depend on the vertical part of :math:`X`.
  
      (This is an alternative implementation.)
  
      Parameters
      ----------
      bundle : PreShapeSpaceBundle
      tangent_vec_a : array-like, shape=[..., k_landmarks, m_ambient]
          Tangent vector at `base_point`.
      tangent_vec_b : array-like, shape=[..., k_landmarks, m_ambient]
          Tangent vector at `base_point`.
      base_point : array-like, shape=[..., k_landmarks, m_ambient]
          Point of the total space.
  
      Returns
      -------
      vector : array-like, shape=[..., k_landmarks, m_ambient]
          Tangent vector at `base_point`, result of the A tensor applied to
          `tangent_vec_a` and `tangent_vec_b`.
  
      References
      ----------
      .. [ONeill]  O’Neill, Barrett. The Fundamental Equations of a
          Submersion, Michigan Mathematical Journal 13, no. 4
          (December 1966): 459–69. https://doi.org/10.1307/mmj/1028999604.
      """
      # Only the horizontal part of a counts
      horizontal_a = bundle.horizontal_projection(
          tangent_vec_a, base_point
      )
      vertical_b, skew = bundle.vertical_projection(
          tangent_vec_b, base_point, return_skew=True
      )
      horizontal_b = tangent_vec_b - vertical_b
  
      # For the horizontal part of b
      transposed_point = Matrices.transpose(base_point)
      sigma = gs.matmul(transposed_point, base_point)
      alignment = gs.matmul(
          Matrices.transpose(horizontal_a), horizontal_b
      )
      right_term = alignment - Matrices.transpose(alignment)
      skew_hor = gs.linalg.solve_sylvester(sigma, sigma, right_term)
      vertical = -gs.matmul(base_point, skew_hor)
  
      # For the vertical part of b
      vert_part = -gs.matmul(horizontal_a, skew)
      tangent_vert = bundle.total_space.to_tangent(
          vert_part, base_point
      )
      horizontal_ = bundle.horizontal_projection(
          tangent_vert, base_point
      )
  
      return vertical + horizontal_
  
imports: import pytest
  import geomstats.backend
  from geomstats.geometry.matrices import Matrices
  from geomstats.test.random import KendalShapeRandomDataGenerator, get_random_times
  from geomstats.test.vectorization import generate_vectorization_data
  from geomstats.test_cases.geometry.base import LevelSetTestCase
  from geomstats.test_cases.geometry.fiber_bundle import FiberBundleTestCase
  from geomstats.test_cases.geometry.quotient_metric import QuotientMetricTestCase
  from geomstats.vectorization import get_batch_shape
label_desc: Use image node_id 1 for calling a global function with example usage: integrability_tensor_alt(bundle, tangent_vec_a, tangent_vec_b, base_point) and returns: unknown