Product Manifolds

Product manifolds 

Definition 

Let \((\mathcal{M}_1,\dots,\mathcal{M}_k)\) be smooth manifolds. Their Cartesian product

\[\mathcal{M} \;\coloneqq\; \mathcal{M}_1 \times \cdots \times \mathcal{M}_k \;=\;\{(x_1,\dots,x_k)\mid x_i\in\mathcal{M}_i\}\]

admits a natural smooth-manifold structure called the product manifold.

Charts and dimension 

If \((U_i,\varphi_i)\) is a chart on \(\mathcal{M}_i\) with \(\varphi_i:U_i\to\mathbb{R}^{n_i}\), then

\[U \;\coloneqq\; U_1\times\cdots\times U_k \subset \mathcal{M},\qquad \varphi \;\coloneqq\; \varphi_1\times\cdots\times\varphi_k\]

defines a chart on \(\mathcal{M}\) via

\[\varphi(x_1,\dots,x_k) \;=\;\big(\varphi_1(x_1),\dots,\varphi_k(x_k)\big)\in\mathbb{R}^{n_1+\cdots+n_k}.\]

Hence,

\[\dim(\mathcal{M}_1\times\cdots\times\mathcal{M}_k) \;=\;\sum_{i=1}^k \dim(\mathcal{M}_i).\]

The canonical projections \(\pi_i:\mathcal{M}\to\mathcal{M}_i\), \(\pi_i(x_1,\dots,x_k)=x_i\), are smooth.

Tangent space: direct sum structure 

At a point \(x=(x_1,\dots,x_k)\in\mathcal{M}\), the tangent space splits as

\[T_x\mathcal{M} \;\cong\; T_{x_1}\mathcal{M}_1 \oplus \cdots \oplus T_{x_k}\mathcal{M}_k.\]

Intuitively, an “infinitesimal motion” on the product is just a tuple of infinitesimal motions on each factor. In coordinates, derivatives and Jacobians often become block-structured, mirroring this direct-sum decomposition.

Riemannian product metric 

If each \(\mathcal{M}_i\) has a Riemannian metric \(\langle\cdot,\cdot\rangle_i\), a common choice on the product is

\[\big\langle (v_1,\dots,v_k),(w_1,\dots,w_k)\big\rangle \;\coloneqq\; \sum_{i=1}^k \langle v_i,w_i\rangle_i, \quad v_i,w_i\in T_{x_i}\mathcal{M}_i.\]

Lie-group special case 

If each \(\mathcal{M}_i\) is a Lie group \(G_i\), then the product \(G=G_1\times\cdots\times G_k\) is a Lie group with component-wise multiplication:

\[(g_1,\dots,g_k)\cdot(h_1,\dots,h_k) \;=\; (g_1h_1,\dots,g_kh_k).\]

Its Lie algebra is the direct sum

\[\mathfrak{g} \;=\; \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_k,\]

and exponential/log maps act componentwise:

\[\exp_G(\xi_1,\dots,\xi_k)=(\exp_{G_1}\xi_1,\dots,\exp_{G_k}\xi_k),\qquad \log_G(g_1,\dots,g_k)=(\log_{G_1}g_1,\dots,\log_{G_k}g_k).\]

In particular, since \(\mathbb{R}^n\) is a Lie group under addition, the common state space

\[SO(3)\times \mathbb{R}^n\]

is (also) a Lie group under \((R,a)\cdot(S,b)=(RS,a+b)\), with Lie algebra \(\mathfrak{so}(3)\oplus\mathbb{R}^n\).

Engineering view: “mixed” state vectors in flight dynamics 

Many GNC/flight-sim states are naturally tuples of heterogeneous components, e.g.

\[x \;=\; (R,\;p,\;v,\;b_g,\;b_a,\;\dots) \in SO(3)\times \mathbb{R}^3\times \mathbb{R}^3\times \mathbb{R}^3\times \mathbb{R}^3\times\cdots.\]

The product-manifold viewpoint says:

You should update each component using the correct geometry.
Euclidean components update by addition; rotational components update by group composition (or a retraction).

A convenient way to formalize this is with “box-plus / box-minus” operators. Let \(\oplus_i\) be the update rule for \(\mathcal{M}_i\) (e.g. \(+\) on \(\mathbb{R}^n\), left-multiplication by \(\exp(\cdot)\) on \(SO(3)\)). Then define for the product:

\[(x_1,\dots,x_k)\;\oplus\;(\delta_1,\dots,\delta_k) \;\coloneqq\; (x_1\oplus_1\delta_1,\;\dots,\;x_k\oplus_k\delta_k),\]

and similarly for \(\ominus\) component-wise.

This is exactly what error-state EKFs and Lie-group / RKMK integrators exploit: propagate on the product manifold, and linearize in the (direct-sum) tangent space.

Why “separate” \(\mathbb{R}^n\) even though it is a Lie group?

You can treat everything as a Lie group if you want (e.g. \(SO(3)\times\mathbb{R}^n\)). In practice we keep \(\mathbb{R}^n\) as “plain Euclidean” because:

its group operation is already \(+\), so the manifold machinery adds no complexity where it is unnecessary;
linearization and covariance bookkeeping are simplest on Euclidean blocks;
non-Euclidean parts (e.g. \(SO(3)\), \(SE(3)\)) are where constraints and curvature matter (e.g. quaternion normalization, avoiding singular charts).

The product-manifold framing cleanly supports both: treat curved components with proper group/retraction updates, and treat Euclidean ones with standard addition.