  We introduce in this paper a manifold optimization framework that utilizes
semi-Riemannian structures on the underlying smooth manifolds. Unlike in
Riemannian geometry, where each tangent space is equipped with a positive
definite inner product, a semi-Riemannian manifold allows the metric tensor to
be indefinite on each tangent space, i.e., possessing both positive and
negative definite subspaces; differential geometric objects such as geodesics
and parallel-transport can be defined on non-degenerate semi-Riemannian
manifolds as well, and can be carefully leveraged to adapt Riemannian
optimization algorithms to the semi-Riemannian setting. In particular, we
discuss the metric independence of manifold optimization algorithms, and
illustrate that the weaker but more general semi-Riemannian geometry often
suffices for the purpose of optimizing smooth functions on smooth manifolds in
practice.
