By the concept of parallel transport, we can transport a vector by Levi-Civita connection over the Riemannian manifold. It allows us to have an isomorphism between tangent spaces along a geodesic and the orthogonal basis transfer along the geodesic.
If we have a quadratic form by a symmetry matrix with some negative and positive eigenvalues, can we transform it via parallel transport along a curve, is not necessarily geodesic? If yes would you please refer me to a suitable reference?
Perhaps it is possible to consider some particular manifolds or connections. But I am not sure if it is possible or not. Thanks for your help!