ManifoldGeometry.project
========================

.. py:method:: t3toolbox.manifold.ManifoldGeometry.project(v)

   .. code-block:: python

      def project(
              self,
              v:  T3Tangent,
      ) -> T3Tangent:  # gauged tangent at v's frame (a DIFFERENT vector)


   The gauge projection ``Pi``: raw cotangent -> Riemannian gradient (orthogonal gauge).

   Orthogonally projects ``v``'s variations onto the gauged tangent space (the gauge conditions
   (48)-(49), Appendix A.3). Represents a DIFFERENT tangent vector than ``v``. This is the map
   that turns a bare adjoint ``J^T r`` into the Riemannian gradient. For the vector-preserving
   gauge fix see :py:meth:`project_oblique`.

   **Safe mode** requires ``v``'s frame to be **orthogonal** (the precondition for this to be the
   orthogonal-gauge projection ``Pi``); it raises otherwise. Skipped under ``safety.unsafe()`` / a
   jax trace.