UniformManifoldGeometry.inner
=============================

.. py:method:: t3toolbox.uniform_manifold.UniformManifoldGeometry.inner(t1, t2)

   .. code-block:: python

      def inner(
              self,
              t1: UT3Tangent,
              t2: UT3Tangent,
      ) -> NDArray:  # the Hilbert-Schmidt inner product, shape = stack_shape (K + C)


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

   The corewise (coordinate) dot, which equals HS on this geometry's orthonormal, gauged frame
   (vectorized over the stack -> shape ``K + C``). 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
   (per stack element, reduced with ``.all()``; 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:`UT3Tangent.corewise_inner`.