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We provide an explicit formula for the Levi-Civita connection and RiemannianHessian for a Riemannian manifold that is a quotient of a manifold embedded inan inner product space with a non-constant metric function. Together with aclassical formula for projection, this allows us to evaluate Riemanniangradient and Hessian for several families of metrics on classical manifolds,including a family of metrics on Stiefel manifolds connecting both the constantand canonical ambient metrics with closed-form geodesics. Using these formulas,we derive Riemannian optimization frameworks on quotients of Stiefel manifolds,including flag manifolds, and a new family of complete quotient metrics on themanifold of positive-semidefinite matrices of fixed rank, considered as aquotient of a product of Stiefel and positive-definite matrix manifold withaffine-invariant metrics. The method is procedural, and in many instances, theRiemannian gradient and Hessian formulas could be derived by symbolic calculus.The method extends the list of potential metrics that could be used in manifoldoptimization and machine learning.

Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics