# Riemannian Geometry with differentiable ambient space and metric operator

arXiv: https://arxiv.org/abs/2105.01583 · HTML: https://mlsanity.com/paper/2105.01583/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 40 references that are in the map.
- First posted on arXiv 4 May 2021.

## Builds on (most cited first, 4 of 4)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics](https://mlsanity.com/paper/2009.10159/) (2020)

## Built on by (most cited first, 1 of 1)

- [Curvatures of Stiefel manifolds with deformation metrics](https://mlsanity.com/paper/2105.01834/) (2021)

## Abstract (arXiv)

We show Riemannian geometry could be studied by identifying the tangent bundle of a Riemannian manifold M with a subbundle of the trivial bundle M × E, obtained by embedding M differentiably in a Euclidean space E. Given such an embedding, we can extend the metric tensor on M to a (positive-definite) operator-valued function acting on E, giving us an embedded ambient structure. The formulas for the Christoffel symbols and Riemannian curvature in local coordinates have simple generalizations to this setup. For a Riemannian submersion mathfrakq:M→ B from an embedded manifold Msubset E, we define a submersed ambient structure and obtain similar formulas, with the O'Neil tensor expressed in terms of the projection to the horizontal bundle HM. Using this framework, we provide the embedded and submersed ambient structures for the double tangent bundle TTM and the tangent of the horizontal bundle THM, describe the fibration of a horizontal bundle over the tangent bundle of the base manifold and extend the notion of a canonical flip to the submersion case. We obtain a formula for horizontal lifts of Jacobi fields, and a new closed-form formula for Jacobi fields of naturally reductive homogeneous spaces. We construct natural metrics on these double tangent bundles, in particular, extending Sasaki and other natural metrics to the submersion case. We illustrate by providing explicit calculations for several manifolds.