---
title: Constraint Manifolds with Corners
url: https://www.emergentmind.com/topics/constraint-manifolds-with-corners
type: topic
---

# Constraint Manifolds with Corners

Searching arXiv for the cited papers to ground the article in published work.
Constraint manifolds with corners are feasible sets defined by equality and inequality constraints whose local geometry is not, in general, that of an ordinary smooth manifold. In the Euclidean optimization setting, they are sets of the form
$$
M=\{x\in\mathbb R^N:h(x)=0,\ g(x)\ge 0\},
$$
with boundary faces and corners at points where one or more inequalities are active [2605.20796]. In a coordinate-free formulation, the same structure appears as the preimage \(X=F^{-1}(\mathcal K)\) of a submanifold with corners \(\mathcal K\subset \mathcal N\) under a \(C^2\) map \(F:\mathcal M\to\mathcal N\) [2110.04882]. In logarithmic differential geometry, manifolds with corners are recovered as positive log differentiable spaces that are log smooth and locally isomorphic to \([0,\infty)^k\times\mathbb R^{n-k}\) [1507.06752]. In Hamiltonian gauge theory with corners, related constraint sets arise as coisotropic submanifolds equipped with boundary flux data, reduction by stages, and corner Poisson geometry [2207.00568].

## 1. Local models and defining structures

In the Euclidean formulation, let \(x\in\mathbb R^N\), let \(h(x)\in\mathbb R^p\) be a vector of smooth equality-constraint functions, and let \(g(x)\in\mathbb R^q\) be a vector of smooth inequality-constraint functions. The feasible set is
$$
M = \{ x\in\mathbb R^N : h(x)=0 ,\   g(x)\ge 0 \}.
$$
The active-constraint index set at \(x\) is
$$
\mathcal A_x=\{i:g_i(x)=0\},\qquad \mathcal I_x=\{i:g_i(x)>0\}.
$$
Under the regularity assumption
$$
\operatorname{rank}\,[\nabla h(x)\ \ \nabla g_{(\mathcal A_x)}(x)] = p+|\mathcal A_x|,
$$
\(M\) is locally a smooth \(p+|\mathcal A_x|\)-codimensional subset of \(\mathbb R^N\), but if \(|\mathcal A_x|>0\) it acquires boundary and corner structure [2605.20796].

The local models separate interior points from corner points. When \(|\mathcal A_x|=0\), one recovers a standard smooth manifold
$$
\overline M=\{h(x)=0\}
$$
of dimension \(n=N-p\). When \(|\mathcal A_x|=m>0\), \(M\) is locally homeomorphic to
$$
H_m^n=\{\theta\in\mathbb R^n:\theta_1,\ldots,\theta_m\ge 0\},
$$
with \(n=N-p\), and a local parametrization \(\phi(\theta)\) provides the corresponding corner chart [2605.20796].

The coordinate-free analogue replaces \(\mathbb R^N\) by a manifold \(\mathcal M\), the target by a manifold \(\mathcal N\), and the Euclidean inequality region by a submanifold with corners \(\mathcal K\subset\mathcal N\). A subset \(\mathcal K\subset\mathcal N\) is a \(k\)-dimensional submanifold with corners if for every \(q\in\mathcal K\) there exist a chart \((\mathcal U,\psi)\) about \(q\), an index \(0\le \ell\le k\), and a surjective linear map \(A:\mathbb R^k\times\{0\}^{n-k}\to\mathbb R^\ell\) such that
$$
\psi(\mathcal K\cap\mathcal U)=\{x\in \psi(\mathcal U)\cap(\mathbb R^k\times\{0\}^{n-k})\mid A\,x\le 0\},
$$
or equivalently
$$
\psi(\mathcal K\cap\mathcal U)=\{x\in\psi(\mathcal U)\mid A\,x\le 0,\ W\,x=0\},
$$
where \(W(x_1,\ldots,x_n)=(x_{k+1},\ldots,x_n)^T\). The feasible set then becomes
$$
X=F^{-1}(\mathcal K)=\{p\in\mathcal M\mid F(p)\in\mathcal K\}
$$
[2110.04882].

In logarithmic differential geometry, the basic local charts are the standard models \(R(P)\) and \(R^+(P)\) attached to a fine monoid \(P\). In particular,
$$
R^+(\mathbb N^k)=[0,\infty)^k,\qquad R(\mathbb N^k)=\mathbb R^k.
$$
A manifold with corners is precisely a positive log differentiable space \(X\) which is log smooth and whose log structure \(M_X\) is everywhere free of rank \(k\), hence locally isomorphic to the chart \(R^+(\mathbb N^k)\times\mathbb R^{n-k}\), equivalently
$$
U\simeq [0,\infty)^k\times \mathbb R^{n-k}.
$$
This embeds the usual coordinate model of corners into a broader theory of log spaces [1507.06752].

## 2. Tangent objects, inner tangents, and local linearization

A central feature of corners is that the tangent object at a boundary or corner point is typically a cone rather than a vector space. In the Euclidean constrained setting, by following all smooth curves \(c(t)\subset M\) with \(c(0)=x\), one obtains
$$
T_xM=\{v\in\mathbb R^N:\nabla h(x)\,v=0,\ \nabla g_{(\mathcal A_x)}(x)\,v\ge 0\}.
$$
When \(|\mathcal A_x|=0\), this is a linear space of dimension \(n\); when \(|\mathcal A_x|>0\), it is a closed polyhedral cone. If \(B\in\mathbb R^{N\times n}\) has orthonormal columns spanning the nullspace of \(\nabla h(x)^T\), then
$$
T_xM=\{v=B\theta:\theta\in\Omega\},
$$
where
$$
\Omega=\{\theta\in\mathbb R^n:\nabla g_{(\mathcal A_x)}(x)\,B\,\theta\ge 0\}\subset\mathbb R^n.
$$
This basis representation isolates the equality constraints in \(B\) and the active inequalities in the coefficient cone \(\Omega\) [2605.20796].

For submanifolds with corners in a manifold \(\mathcal N\), the adapted chart description yields three related objects at a point \(q\in\mathcal K\). The tangent space \(T_q\mathcal K\) is the linear subspace represented by vectors \(x\in\mathbb R^n\) satisfying \(W\,x=0\). The cone of inner tangents is
$$
T^i_q\mathcal K=\{v\in T_q\mathcal K\mid A\,v\le 0\},
$$
a closed convex polyhedral cone in \(T_q\mathcal N\). The zero-tangent subspace is
$$
T^0_q\mathcal K=\{v\in T_q\mathcal K\mid A\,v=0\},
$$
which is the lineality space of \(T^i_q\mathcal K\) [2110.04882].

For feasible sets \(X=F^{-1}(\mathcal K)\), the linearizing cone at \(p\in X\), with \(q=F(p)\), is
$$
L_pX=\{v\in T_p\mathcal M\mid F'(p)\,v\in T^i_q\mathcal K\}.
$$
Under the Zowe-Kurcyusz constraint qualification
$$
\operatorname{Im}F'(p)-T^i_q\mathcal K=T_q\mathcal N \tag{ZKRCQ}
$$
one has \(T_pX=L_pX\). This gives a coordinate-free analogue of the Euclidean tangent-cone formula and identifies the correct first-order feasible directions when corners are present [2110.04882].

A common misconception is that the presence of inequalities merely adds boundary points to an otherwise ordinary manifold. The formulas above show a sharper picture: once active inequalities appear, the relevant first-order object is not generally a linear tangent space but a convex cone. This is explicit both in the Euclidean formula for \(T_xM\) and in the inner-tangent cone \(T^i_q\mathcal K\).

## 3. Optimization on constraint manifolds with corners

The optimization framework in CMC-Opt endows \(M\) with the metric induced by the ambient Euclidean inner product,
$$
\langle u,v\rangle_x=u^Tv.
$$
For any ambient vector \(w\in\mathbb R^N\), projection onto the tangent cone is defined by the quadratic program
$$
\theta^*=\arg\min_{\theta\in\Omega}\|B\theta-w\|^2,\qquad P_x(w)=B\theta^*.
$$
Retraction from \(x\) along \(v\in T_xM\) is given by
$$
R_x(v)=\arg\min_{y\in\mathbb R^N}\|y-(x+v)\|^2\quad\text{s.t.}\quad h(y)=0,\ g(y)\ge 0.
$$
It satisfies \(R_x(0)=x\) and \(DR_x(0)=I\). These two constructions replace the standard tangent-space projection and manifold retraction of smooth Riemannian optimization by cone-aware analogues [2605.20796].

If \(F:M\to\mathbb R\) is the cost and \(f:\mathbb R^N\to\mathbb R\) is a smooth extension, then for \(v\in T_xM\),
$$
D\,F(x)[v]=\nabla f(x)^T v.
$$
The Riemannian gradient is the tangent-cone projection of the ambient gradient,
$$
\operatorname{grad}F(x)=P_x(\nabla f(x))=B\theta^*,
$$
where \(\theta^*\) solves
$$
\theta^*=\arg\min_{\theta\in\Omega}\|B\theta-\nabla f(x)\|^2.
$$
For second-order models, one may form
$$
H=B^T\nabla^2 f(x)\,B
$$
and solve the trust-region subproblem
$$
\min_{\theta\in\Omega,\ \|\theta\|\le \Delta}\ \frac12\,\theta^T H\theta+(B^T\nabla f)^T\theta.
$$
This yields a Newton-like direction respecting all active inequalities [2605.20796].

The active set is updated dynamically:
$$
\mathcal A_x=\{i:g_i(x)\approx 0\}.
$$
When an inequality becomes inactive or newly active, the cone \(\Omega\) gains or loses faces, so that the local chart \(H_m^n\) glues smoothly across corner transitions. In product settings \(\{M_1,\ldots,M_k\}\), the paper gives an explicit Riemannian gradient descent scheme on CMCs with the steps: compute \(\nabla f(x^k)\), form \(B\) and \(\Omega\), solve the projection QP, set \(v^k=-\alpha_k B\theta^*\), and retract \(x^{k+1}=R_{x^k}(v^k)\) [2605.20796].

The paper states that, under mild Lipschitz-gradient and regularity assumptions, one can import standard convergence guarantees for Riemannian gradient and trust-region methods. It also records the following statements: any accumulation point of gradient descent satisfies the first-order Karush-Kuhn-Tucker conditions on \(M\); with a sufficiently small fixed or diminishing step-size the Riemannian gradient descent on a CMC converges to a KKT point; and trust-region-Newton with an exact Hessian and properly chosen radius yields local superlinear convergence around a nondegenerate second-order KKT point [2605.20796]. This suggests that corner-aware manifold methods can be viewed as an extension of standard manifold optimization rather than a departure from it.

The illustrative large-scale kinodynamic planning example is a quadruped jump with 13 rigid-body links, 70 time steps, four contact phases, equality constraints given by joint kinematics, Newton-Euler link dynamics, and contact-stationarity, and inequality constraints given by collision avoidance, Coulomb friction cones, and joint angle/torque limits. The reported quantitative summary is:

- Penalty: Dim. \(32\,194\), Violation \(1.3\mathrm e{+01}\), Cost \(2.7\mathrm e{+05}\)
- AugL: Dim. \(32\,194\), Violation \(7.7\mathrm e{+00}\), Cost \(3.1\mathrm e{+04}\)
- SQP: Dim. \(32\,194\), Violation \(1.0\mathrm e{-01}\), Cost \(1.3\mathrm e{+06}\)
- CM-Opt: Dim. \(2\,260\), Violation \(4.2\mathrm e{-01}\), Cost \(7.0\mathrm e{+05}\)
- CMC-Opt: Dim. \(2\,260\), Violation \(0\), Cost \(709.78\)

The paper further states that the problem size was reduced from \(32\,194\to 2\,260\) by eliminating slack variables, and that only CMC-Opt achieved zero constraint violation and the lowest final cost [2605.20796].

## 4. First- and second-order optimality in the manifold-valued setting

For manifold-valued constraints, the feasible set is
$$
X=F^{-1}(\mathcal K),
$$
with \(F\in C^2(\mathcal M,\mathcal N)\) and \(\mathcal K\subset\mathcal N\) a submanifold with corners. Locally in charts \(p\mapsto x\in\mathbb R^m\), \(q=F(p)\mapsto y\in\mathbb R^n\), and adapted \(\psi\) on \(\mathcal N\), the feasible set takes the form
$$
\{x\in\mathbb R^m\mid A\,g(x)\le 0,\ W\,g(x)=0\},\qquad g=\psi\circ F\circ\varphi^{-1}.
$$
This reduces the geometric problem to a convex-cone constrained nonlinear program in local coordinates without changing the intrinsic objects [2110.04882].

Let \(f\in C^2(\mathcal M,\mathbb R)\) and let \(p_*\in X\) be a local minimizer satisfying ZKRCQ. Then there exists a multiplier \(\mu\in T_q^*\mathcal N\), with \(q=F(p_*)\), such that

1. Stationarity:
$$
df(p_*)+\mu\circ F'(p_*)=0\quad\text{in }T^*_{p_*}\mathcal M,
$$

2. Complementarity:
$$
\mu\in (T^i_q\mathcal K)^\circ:=\{\lambda\in T_q^*\mathcal N\mid \langle\lambda,w\rangle\le 0\ \forall\,w\in T^i_q\mathcal K\}.
$$

The set of all such multipliers is nonempty, compact, and a singleton under the stronger LICQ condition
$$
\operatorname{Im}F'(p_*)-T^0_q\mathcal K=T_q\mathcal N.
$$
In an adapted chart at \(q\), the multiplier has the representation
$$
\mu_\psi=
\begin{pmatrix}
A^T\lambda_I\\
W^T\lambda_E
\end{pmatrix},
\qquad
\lambda_I\in\mathbb R^\ell_{\ge 0},\ \lambda_E\in\mathbb R^{n-k},
$$
and stationarity becomes
$$
D f_\phi(x_*)^T + Dg(x_*)^T\bigl(A^T\lambda_I+W^T\lambda_E\bigr)=0,\qquad \lambda_I\ge 0.
$$
These formulas generalize the familiar KKT system to manifold-valued constraints with corner targets [2110.04882].

Second-order analysis uses the critical cone
$$
C_{p_*}=\{v\in T_{p_*}\mathcal M\mid F'(p_*)v\in T^i_q\mathcal K,\ df(p_*)\,v=0\}
$$
and a Hessian of the Lagrangian. If \(h\) is chosen so that \(dh(q)=\mu\), then for
$$
L(p)=f(p)+h(F(p)),
$$
the bilinear form
$$
d^2L(p_*,\mu):T_{p_*}\mathcal M\times T_{p_*}\mathcal M\to\mathbb R
$$
is well defined and independent of local extensions of \(\mu\). The second-order conditions are:

- Necessary under LICQ:
$$
d^2L(p_*,\mu)[v,v]\ge 0\qquad \forall\,v\in C_{p_*},
$$

- Sufficient under ZKRCQ plus strictness:
$$
d^2L(p_*,\mu)[v,v]>0\qquad \forall\,v\in C_{p_*}\setminus\{0\}.
$$

The paper emphasizes invariance: the tangent cone \(T_pX\), dual cone \((T^i_q\mathcal K)^\circ\), Lagrange multipliers \(\mu\), and the Hessian \(d^2L\) are invariant under change of adapted local chart on \(\mathcal N\) and under equivalent reformulations \(F\mapsto \Lambda\circ F\) where \(\Lambda\) preserves \(\mathcal K\) [2110.04882].

## 5. Log differentiable spaces, fans, and resolution of singular corners

A log differentiable space is a locally ringed space \(X\) over \(\mathbb R\) equipped with a log structure
$$
\alpha:M_X\to\mathcal O_X
$$
where \(M_X\) is a sheaf of commutative monoids and \(\alpha\) induces an isomorphism
$$
\alpha|_{\alpha^{-1}(\mathcal O_X^\times)}:\alpha^{-1}(\mathcal O_X^\times)\simeq \mathcal O_X^\times.
$$
Equivalently, one works with the submonoid \(\mathcal O_X^+\subset \mathcal O_X\) of non-negative functions and a map
$$
\alpha^+:M_X\to\mathcal O_X^+.
$$
In this language, a manifold with corners is precisely a positive log differentiable space which is log smooth and locally modeled on \(R^+(\mathbb N^k)\times\mathbb R^{n-k}\) [1507.06752].

On \([0,\infty)^k\), the chart monoid homomorphism is
$$
h:\mathbb N^k\to\mathcal O_X^+(U),\qquad e_i\mapsto x_i,
$$
where \(x_i\) is the \(i\)-th coordinate function. After taking the associated log structure, one obtains a sheaf of monoids \(M_X\) with a map \(\alpha^+:M_X\to\mathcal O_X^+\). The characteristic monoid at a point \(p\) is
$$
\overline M_{X,p}=M_{X,p}/M_{X,p}^\times,
$$
naturally isomorphic to \(\mathbb N^k\), and on stalks one has the exact sequence
$$
0\to \mathcal O_{X,p}^\times\to M_{X,p}\to \overline M_{X,p}\to 0,
$$
which is split on manifolds with corners by taking the chart [1507.06752].

The paper’s chart criterion for log smoothness states that a morphism \(f:X\to Y\) of fine log differentiable spaces is log smooth if locally on \(X\) there exist charts
$$
P\to M_X(U),\qquad Q\to M_Y(V),
$$
together with a map of monoids \(Q\to P\) making
$$
X|_U\to V\times_{R(Q)}R(P)
$$
a classically smooth map of differentiable spaces. Log smooth morphisms are stable under composition, base-change, and are local in \(X\) and \(Y\). In particular, the projection
$$
[0,\infty)^k\times\mathbb R^{n-k}\to \mathbb R^{n-k}
$$
is log smooth [1507.06752].

The combinatorics of corners are encoded by fans. A fan is a sharp locally monoidal space locally isomorphic to \(\operatorname{Spec}P\) for some fine monoid \(P\). Its points are prime ideals, equivalently faces, of \(P\), and the structure sheaf is the localization presheaf. In the corner-model case \(P=\mathbb N^k\), the faces are the coordinate hyperplanes. The “boundary” functor corresponds on fans to taking the disjoint union of the maximal proper faces [1507.06752].

The resolution theory is formulated monoidally and then realized differentiably. For every fs monoid \(P\) there is a canonical combinatorial sequence of blowups of faces
$$
I(P)=(I_1,I_2,\ldots,I_m)
$$
on \(\operatorname{Spec}P\), functorial in \(P\), so that the final blowup \(\operatorname{Spec}P'\) is free, equivalently \(P'\simeq \mathbb N^k\). Passing to log differentiable spaces via \(R^+(-)\), these blowups pull back to Euclidean-proper, locally-projective, log smooth maps
$$
R^+(P')\to R^+(P),
$$
which over the smooth locus are isomorphisms. More generally, if \(X\) is any fs log differentiable space, one applies the functorial blowup on the associated fan \(X^\sharp\) to obtain a free fan \(F\to X^\sharp\); the terminal object in the Kato-category lifts this subdivision to a log smooth, Euclidean-proper, surjective map of PLDS
$$
X'\to X
$$
with \(X'\) free, that is, a genuine manifold with corners. Over the locus where \(X\) was already free, the blowup is an isomorphism [1507.06752].

The examples in the paper include a linear inequality region in \(\mathbb R^n\), such as \(\{x:x_i\ge 0,\ \Sigma x_i\ge 1\}\), feasible sets of polynomial inequalities with “nice” log-smooth defining equations, and a bounded intersection of half-spaces becoming a compact log smooth PLDS whose resolution is the usual manifold with corners of the polytope. This suggests a direct connection between corner singularities in constrained systems and functorial toric-style desingularization.

## 6. Constraint reduction, flux superselection, and corners in gauge theory

In Hamiltonian gauge theory with corners, the bulk manifold is a compact manifold with boundary \(\Sigma\), with corner \(\partial\Sigma\), and the space of fields is
$$
P=\Gamma(\Sigma,E)
$$
equipped with a local symplectic form \(\omega=\int_\Sigma \underline{\omega}\), acted on by a local gauge group \(G\) with Lie algebra \(\mathfrak g\). The local momentum form \(H\in\underline{\Omega}^{\mathrm{top},0}(\Sigma\times P;\mathfrak g^*)\) is defined by
$$
i_{\rho(\xi)}\underline{\omega}=d\langle H,\xi\rangle,\qquad \forall \xi\in\mathfrak g,
$$
with integrated map
$$
\mu_0:=\int_\Sigma H\in C^\infty(P;\mathfrak g^*).
$$
By the Takens-Zuckerman decomposition,
$$
H=\underline H+d\underline h,
$$
with \(\underline H\) of order \(0\) and exact term \(d\underline h\), so that
$$
\mu_0(A)=\int_\Sigma \underline H,\qquad \mu_\partial(A)=\int_\Sigma d\underline h=\int_{\partial\Sigma}\underline h.
$$
Here \(\mu_0\) is equivariant for the action of the constraint gauge subgroup \(G_0\triangleleft G\), while \(\mu_\partial\) is a momentum map, up to cocycle, for the residual action of \(G/G_0\) [2207.00568].

The constraint set is
$$
C:=\mu_0^{-1}(0)\subset P.
$$
The paper proves that \(C\) is a coisotropic submanifold in \((P,\omega)\):
$$
TC^\omega=\rho(C\times\mathfrak g_0)\subset TC.
$$
Equivalently, in the theorem stated in the details,
$$
T_\phi C^\omega=\rho_\phi(\mathfrak g_0).
$$
The ideal \(\mathfrak g_0=\operatorname{Ann}(F,\mathfrak g)\), where \(F=\operatorname{Im}(\mu_\partial)\subset \mathfrak g^*\), is the maximal ideal for which \(\mu_0\) factors through \(\mathfrak g_0\) and \(\mu_0^{-1}(0)=C\) [2207.00568].

Reduction proceeds by stages. The first stage is constraint reduction at \(\mu_0=0\) by \(G_0\):
$$
(P,\omega)\mathbin{//}_0 G_0:=\mu_0^{-1}(0)/G_0,
$$
with
$$
\pi_0^*\omega_0=i_C^*\omega,\qquad \omega_0\in\Omega^2(C/G_0).
$$
Equivalently,
$$
(C/C^\omega)\simeq \mu_0^{-1}(0)/G_0.
$$
The second stage is flux superselection by \(G/G_0\). For each coadjoint orbit \(\mathcal O_f\subset (\mathfrak g/\mathfrak g_0)^*\), define
$$
S_{[f]}:=(\mu_\partial)^{-1}(\mathcal O_f)\subset C/G_0,\qquad
U_{[f]}:=S_{[f]}/(G/G_0).
$$
Point reduction at \(f\in (\mathfrak g/\mathfrak g_0)^*\) yields
$$
\mu_\partial^{-1}(f)/(G/G_0)_f\simeq U_{[f]}.
$$
The disjoint union
$$
U_C=\bigsqcup_{[f]}U_{[f]}
$$
carries a natural Poisson structure whose symplectic leaves are exactly the \(U_{[f],q}\) [2207.00568].

Corner data are encoded by an action Lie algebroid restricted to \(\partial\Sigma\):
$$
A_\partial=P_\partial\times (\mathfrak g/\mathfrak g_0)\to P_\partial,\qquad
P_\partial=\Gamma(\partial\Sigma,E|_{\partial\Sigma}).
$$
The \(2\)-form
$$
\omega_\partial=\langle d_H\mu_\partial,\ d_V\bar\xi\rangle
$$
is weakly symplectic and basic for the projection to \(P_\partial\). Hence \(A_\partial\to P_\partial\) becomes a symplectic Lie algebroid, and its anchor \(\rho_\partial:A_\partial\to TP_\partial\) induces on \(P_\partial\) a partial Poisson bivector
$$
\Pi_\partial^\sharp=\rho_\partial:T^pP_\partial\to TP_\partial.
$$
Both the on-shell corner data \(C_\partial\hookrightarrow P_\partial\) and the fully reduced \(C/G=U_C\) project onto the same base
$$
\mathcal B:=C_\partial/(G/G_0),
$$
which parametrizes flux superselection sectors in both constructions [2207.00568].

For Yang-Mills on \((\Sigma,\partial\Sigma)\), with \(G\) compact or semisimple, \(P=A\times E\), \(E\in\Omega^{n-1}(\Sigma,\mathfrak g^*)\), \(A\in\Omega^1(\Sigma,\mathfrak g)\), and
$$
\underline\omega=\operatorname{tr}(dE\wedge dA),
$$
the constraint form is \(\langle H,\xi\rangle=\operatorname{tr}((d_AE)\xi)\), the flux form satisfies \(dh=-d\,\operatorname{tr}(E\xi)\), and
$$
\mathfrak g_0=\{\xi\mid \xi|_{\partial\Sigma}=0\},\qquad
G_0=\{g\mid g|_{\partial\Sigma}=1\},\qquad
G/G_0\simeq \operatorname{Map}(\partial\Sigma,G).
$$
The constraint set is
$$
C=\{d_AE=0\},
$$
the first reduction is
$$
C/G_0\simeq T(A/G_0)\times F,
$$
and the second reduction at orbit \(\mathcal O_f\subset F\) yields
$$
U_{[f]}\simeq T(A/G_0)\times \mathcal O_f.
$$
Hence
$$
U_C\simeq \bigsqcup_{[f]}T(A/G_0)\times \mathcal O_f,
$$
a Weinstein bundle [2207.00568].

Across these settings, “constraint manifolds with corners” do not refer to a single formalism. The phrase names a family of closely related structures: feasible sets with active inequalities in optimization, preimages of cornered targets in manifold-valued analysis, positive log differentiable spaces in logarithmic geometry, and coisotropic constraint sets with boundary flux data in gauge theory. The common theme is that corners require tangent cones, face combinatorics, and reduction or resolution procedures that are not captured by the theory of smooth manifolds alone.

Source: https://www.emergentmind.com/topics/constraint-manifolds-with-corners