ManifoldGeometry.inner
======================

.. py:method:: t3toolbox.manifold.ManifoldGeometry.inner(t1, t2)

   .. code-block:: python

      def inner(
              self,
              t1: T3Tangent,
              t2: T3Tangent,
      ) -> NDArray:  # the Hilbert-Schmidt inner product, shape = stack_shape (K + C)


   The **Hilbert-Schmidt** inner product of two tangents -- the Riemannian metric on ``M``.

   Computes the corewise (coordinate) dot, which equals HS on this geometry's orthonormal, gauged
   frame. In **safe mode** it checks the preconditions for that equality: the two tangents share a
   frame, the frame is orthogonal, and **both** variations are gauged (minimal rank is a documented
   caveat -- see ``docs/numerical_contracts.md``). For the raw coordinate dot with no HS claim
   and no orthogonal/gauge check, use :py:meth:`T3Tangent.corewise_inner`.