Retractions and tangent charts

Estimation moves points. On $\R^\dimension$ a step is an addition, $\point + \tangent$; on a curved manifold, such as the rotations or the unit sphere, there is no addition, and a retraction takes its place. This note defines it, shows how it turns every tangent space into a chart, and introduces the notation $\oplus$ and $\ominus$ used throughout these notes.

The definition#

Let $\manifold$ be a smooth manifold of dimension $\dimension$. In computation, a tangent space $\T_\point\manifold$ is never handled abstractly: each is identified with $\R^\dimension$ by a basis chosen smoothly in $\point$, and a tangent vector is a column of $\dimension$ numbers. We assume such an identification throughout. Some manifolds have none that is smooth everywhere: by the hairy ball theorem, no smooth choice of tangent bases covers the whole sphere $S^2$. There the identification is smooth on an open part of the manifold, and a computation stays inside it.

The definition is the one used in optimization on manifolds, where retractions replace the exponential map of a Riemannian manifold. The exponential map is a retraction, but so is any map that agrees with it to first order, which is usually cheaper. Nothing here needs a metric.

Two examples. On $\R^\dimension$, $R_\point(\tangent) = \point + \tangent$. On the positive reals, with $\T_\point(0, \infty)$ identified with $\R$ by measuring relative change, $R_\point(\tangent) = \point\,e^{\tangent}$: it is defined for every $\tangent$, never leaves the positive reals, and satisfies $\frac{\dd}{\dd\tangent} \point e^\tangent \big|_{0} = \point$, which is the identity in the chosen identification (a tangent at $\point$ of coordinate $1$ is the velocity $\point$).

Tangent charts#

Because $\D R_\point(0)$ is invertible, the inverse function theorem makes $R_\point$ a diffeomorphism from a neighbourhood of $0 \in \R^\dimension$ onto a neighbourhood of $\point$. So $R_\point$ is a chart of $\manifold$ around $\point$, the tangent chart at $\point$, and every point near $\point$ has coordinates in it. As is common in robotics, we write

the second for $\otherPoint$ in the neighbourhood where $R_\point$ is invertible. In words, $\otherPoint \ominus \point$ is where $\otherPoint$ sits in the chart at $\point$. By construction,

for every $\tangent$ in the chart’s domain, and $\point \oplus (\otherPoint \ominus \point) = \otherPoint$ for every $\otherPoint$ in its image. On $\R^\dimension$, $\oplus$ and $\ominus$ are addition and subtraction; on the positive reals above, $\otherPoint \ominus \point = \log(\otherPoint / \point)$.

The mirror identity#

On $\R^\dimension$, $\point \ominus (\point \oplus \tangent) = -\tangent$ as well: the step back from $\point \oplus \tangent$ to $\point$ is the reverse of the step forward. On a curved manifold it generally is not, because the left side is measured in the chart at the moved point $\point \oplus \tangent$, not at $\point$.

The identity holds exactly in some cases: on $\R^\dimension$, on the positive reals above, and on any Lie group whose retraction is $R_\point(\tangent) = \point \operatorname{Exp}(\tangent)$, since then $\point \ominus (\point \oplus \tangent) = \operatorname{Log}(\operatorname{Exp}(\tangent)^{-1}) = -\tangent$. It fails at second order for other retractions, and a computation that needs “the point whose coordinate is $-\tangent$” must not assume it.

A manifold whose retraction is $\point + \tangent$, so that both identities hold exactly and every chart is the same affine chart, is called flat in these notes.