Product manifolds

Several quantities taken together, such as a pose and a velocity, or every state in a window of an estimator, are one point of a product manifold. This note shows that the product inherits a retraction from its factors, and how derivatives in its charts break into blocks.

The product and its tangent space#

Let $\manifold_1, \dots, \manifold_k$ be manifolds of dimensions $\dimension_1, \dots, \dimension_k$, each with a retraction. Their product $\manifold = \manifold_1 \times \cdots \times \manifold_k$ is a manifold of dimension $\dimension = \sum_i \dimension_i$, whose points are tuples $\point = (\point_1, \dots, \point_k)$. Its tangent space at $\point$ is the direct sum of the factors’, identified with $\R^\dimension$ by stacking:

(The $\oplus$ of vector spaces here is the direct sum, not the retraction.)

Derivatives in product charts#

Let $\map : \manifold \to \otherManifold$ be smooth, with $\otherPoint = \map(\point)$. Because a step in the product chart moves each factor in its own chart, the derivative in charts of $\map$ is a row of blocks,

each the derivative in the $i$-th factor’s chart with the other factors held fixed. A map into a product is a column of blocks in the same way, one per factor of the output. For a scalar $f : \manifold \to \R$, the chart gradient stacks the factors’ partial gradients, $g = (g_1; \dots; g_k)$, and the chart Hessian is the block matrix whose $(i, j)$ block differentiates $g_i$ along the $j$-th factor: $H_{ij}(\point)\, \tangent_j = \frac{\dd}{\dd t} g_i(\point_1, \dots, \point_j \oplus t \tangent_j, \dots, \point_k)$ at $t = 0$.

In particular, the retraction of the product is, as a map of the pair $(\point, \tangent)$, the factors’ retractions side by side, and its derivatives of every order are block diagonal in the factors. Nothing about a product’s derivatives requires more than its factors’ derivatives and the bookkeeping of (1).