Constraint Manifolds with Corners

• The paper introduces a unified framework for constraint manifolds with corners, showing how active inequalities transform smooth tangent spaces into convex cones.
• It develops coordinate-free models and cone-aware optimization methods that extend classical Riemannian techniques under standard regularity conditions.
• The study bridges Euclidean optimization, logarithmic geometry, and Hamiltonian gauge theory, with applications such as kinodynamic planning and flux reduction.

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 (Zhang et al., 20 May 2026). 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 $F:\mathcal M\to\mathcal N$ (Bergmann et al., 2021). In logarithmic differential geometry, manifolds with corners are recovered as positive log differentiable spaces that are log smooth and locally isomorphic to $[0,\infty)^k\times\mathbb R^{n-k}$ (Gillam et al., 2015). 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 (Riello et al., 2022).

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=F^{-1}(\mathcal K)$0 is

$X=F^{-1}(\mathcal K)$1

Under the regularity assumption

$X=F^{-1}(\mathcal K)$2

$X=F^{-1}(\mathcal K)$3 is locally a smooth $X=F^{-1}(\mathcal K)$4-codimensional subset of $X=F^{-1}(\mathcal K)$5, but if $X=F^{-1}(\mathcal K)$6 it acquires boundary and corner structure (Zhang et al., 20 May 2026).

The local models separate interior points from corner points. When $X=F^{-1}(\mathcal K)$7, one recovers a standard smooth manifold

$X=F^{-1}(\mathcal K)$8

of dimension $X=F^{-1}(\mathcal K)$9. When $\mathcal K\subset \mathcal N$0, $\mathcal K\subset \mathcal N$1 is locally homeomorphic to

$\mathcal K\subset \mathcal N$2

with $\mathcal K\subset \mathcal N$3, and a local parametrization $\mathcal K\subset \mathcal N$4 provides the corresponding corner chart (Zhang et al., 20 May 2026).

The coordinate-free analogue replaces $\mathcal K\subset \mathcal N$5 by a manifold $\mathcal K\subset \mathcal N$6, the target by a manifold $\mathcal K\subset \mathcal N$7, and the Euclidean inequality region by a submanifold with corners $\mathcal K\subset \mathcal N$8. A subset $\mathcal K\subset \mathcal N$9 is a $C^2$0-dimensional submanifold with corners if for every $C^2$1 there exist a chart $C^2$2 about $C^2$3, an index $C^2$4, and a surjective linear map $C^2$5 such that

$C^2$6

or equivalently

$C^2$7

where $C^2$8. The feasible set then becomes

$C^2$9

(Bergmann et al., 2021).

In logarithmic differential geometry, the basic local charts are the standard models $F:\mathcal M\to\mathcal N$0 and $F:\mathcal M\to\mathcal N$1 attached to a fine monoid $F:\mathcal M\to\mathcal N$2. In particular,

$F:\mathcal M\to\mathcal N$3

A manifold with corners is precisely a positive log differentiable space $F:\mathcal M\to\mathcal N$4 which is log smooth and whose log structure $F:\mathcal M\to\mathcal N$5 is everywhere free of rank $F:\mathcal M\to\mathcal N$6, hence locally isomorphic to the chart $F:\mathcal M\to\mathcal N$7, equivalently

$F:\mathcal M\to\mathcal N$8

This embeds the usual coordinate model of corners into a broader theory of log spaces (Gillam et al., 2015).

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 $F:\mathcal M\to\mathcal N$9 with $[0,\infty)^k\times\mathbb R^{n-k}$0, one obtains

$[0,\infty)^k\times\mathbb R^{n-k}$1

When $[0,\infty)^k\times\mathbb R^{n-k}$2, this is a linear space of dimension $[0,\infty)^k\times\mathbb R^{n-k}$3; when $[0,\infty)^k\times\mathbb R^{n-k}$4, it is a closed polyhedral cone. If $[0,\infty)^k\times\mathbb R^{n-k}$5 has orthonormal columns spanning the nullspace of $[0,\infty)^k\times\mathbb R^{n-k}$6, then

$[0,\infty)^k\times\mathbb R^{n-k}$7

where

$[0,\infty)^k\times\mathbb R^{n-k}$8

This basis representation isolates the equality constraints in $[0,\infty)^k\times\mathbb R^{n-k}$9 and the active inequalities in the coefficient cone $x\in\mathbb R^N$0 (Zhang et al., 20 May 2026).

For submanifolds with corners in a manifold $x\in\mathbb R^N$1, the adapted chart description yields three related objects at a point $x\in\mathbb R^N$2. The tangent space $x\in\mathbb R^N$3 is the linear subspace represented by vectors $x\in\mathbb R^N$4 satisfying $x\in\mathbb R^N$5. The cone of inner tangents is

$x\in\mathbb R^N$6

a closed convex polyhedral cone in $x\in\mathbb R^N$7. The zero-tangent subspace is

$x\in\mathbb R^N$8

which is the lineality space of $x\in\mathbb R^N$9 (Bergmann et al., 2021).

For feasible sets $h(x)\in\mathbb R^p$0, the linearizing cone at $h(x)\in\mathbb R^p$1, with $h(x)\in\mathbb R^p$2, is

$h(x)\in\mathbb R^p$3

Under the Zowe-Kurcyusz constraint qualification

$h(x)\in\mathbb R^p$4

one has $h(x)\in\mathbb R^p$5. This gives a coordinate-free analogue of the Euclidean tangent-cone formula and identifies the correct first-order feasible directions when corners are present (Bergmann et al., 2021).

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 $h(x)\in\mathbb R^p$6 and in the inner-tangent cone $h(x)\in\mathbb R^p$7.

3. Optimization on constraint manifolds with corners

The optimization framework in CMC-Opt endows $h(x)\in\mathbb R^p$8 with the metric induced by the ambient Euclidean inner product,

$h(x)\in\mathbb R^p$9

For any ambient vector $g(x)\in\mathbb R^q$0, projection onto the tangent cone is defined by the quadratic program

$g(x)\in\mathbb R^q$1

Retraction from $g(x)\in\mathbb R^q$2 along $g(x)\in\mathbb R^q$3 is given by

$g(x)\in\mathbb R^q$4

It satisfies $g(x)\in\mathbb R^q$5 and $g(x)\in\mathbb R^q$6. These two constructions replace the standard tangent-space projection and manifold retraction of smooth Riemannian optimization by cone-aware analogues (Zhang et al., 20 May 2026).

If $g(x)\in\mathbb R^q$7 is the cost and $g(x)\in\mathbb R^q$8 is a smooth extension, then for $g(x)\in\mathbb R^q$9,

$M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$0

The Riemannian gradient is the tangent-cone projection of the ambient gradient,

$M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$1

where $M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$2 solves

$M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$3

For second-order models, one may form

$M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$4

and solve the trust-region subproblem

$M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$5

This yields a Newton-like direction respecting all active inequalities (Zhang et al., 20 May 2026).

The active set is updated dynamically:

$M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$6

When an inequality becomes inactive or newly active, the cone $M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$7 gains or loses faces, so that the local chart $M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$8 glues smoothly across corner transitions. In product settings $M = \{ x\in\mathbb R^N : h(x)=0 ,\ g(x)\ge 0 \}.$9, the paper gives an explicit Riemannian gradient descent scheme on CMCs with the steps: compute $X=F^{-1}(\mathcal K)$00, form $X=F^{-1}(\mathcal K)$01 and $X=F^{-1}(\mathcal K)$02, solve the projection QP, set $X=F^{-1}(\mathcal K)$03, and retract $X=F^{-1}(\mathcal K)$04 (Zhang et al., 20 May 2026).

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 $X=F^{-1}(\mathcal K)$05; 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 (Zhang et al., 20 May 2026). 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. $X=F^{-1}(\mathcal K)$06, Violation $X=F^{-1}(\mathcal K)$07, Cost $X=F^{-1}(\mathcal K)$08
• AugL: Dim. $X=F^{-1}(\mathcal K)$09, Violation $X=F^{-1}(\mathcal K)$10, Cost $X=F^{-1}(\mathcal K)$11
• SQP: Dim. $X=F^{-1}(\mathcal K)$12, Violation $X=F^{-1}(\mathcal K)$13, Cost $X=F^{-1}(\mathcal K)$14
• CM-Opt: Dim. $X=F^{-1}(\mathcal K)$15, Violation $X=F^{-1}(\mathcal K)$16, Cost $X=F^{-1}(\mathcal K)$17
• CMC-Opt: Dim. $X=F^{-1}(\mathcal K)$18, Violation $X=F^{-1}(\mathcal K)$19, Cost $X=F^{-1}(\mathcal K)$20

The paper further states that the problem size was reduced from $X=F^{-1}(\mathcal K)$21 by eliminating slack variables, and that only CMC-Opt achieved zero constraint violation and the lowest final cost (Zhang et al., 20 May 2026).

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

For manifold-valued constraints, the feasible set is

$X=F^{-1}(\mathcal K)$22

with $X=F^{-1}(\mathcal K)$23 and $X=F^{-1}(\mathcal K)$24 a submanifold with corners. Locally in charts $X=F^{-1}(\mathcal K)$25, $X=F^{-1}(\mathcal K)$26, and adapted $X=F^{-1}(\mathcal K)$27 on $X=F^{-1}(\mathcal K)$28, the feasible set takes the form

$X=F^{-1}(\mathcal K)$29

This reduces the geometric problem to a convex-cone constrained nonlinear program in local coordinates without changing the intrinsic objects (Bergmann et al., 2021).

Let $X=F^{-1}(\mathcal K)$30 and let $X=F^{-1}(\mathcal K)$31 be a local minimizer satisfying ZKRCQ. Then there exists a multiplier $X=F^{-1}(\mathcal K)$32, with $X=F^{-1}(\mathcal K)$33, such that

1. Stationarity:

$X=F^{-1}(\mathcal K)$34

2. Complementarity:

$X=F^{-1}(\mathcal K)$35

The set of all such multipliers is nonempty, compact, and a singleton under the stronger LICQ condition

$X=F^{-1}(\mathcal K)$36

In an adapted chart at $X=F^{-1}(\mathcal K)$37, the multiplier has the representation

$X=F^{-1}(\mathcal K)$38

and stationarity becomes

$X=F^{-1}(\mathcal K)$39

These formulas generalize the familiar KKT system to manifold-valued constraints with corner targets (Bergmann et al., 2021).

Second-order analysis uses the critical cone

$X=F^{-1}(\mathcal K)$40

and a Hessian of the Lagrangian. If $X=F^{-1}(\mathcal K)$41 is chosen so that $X=F^{-1}(\mathcal K)$42, then for

$X=F^{-1}(\mathcal K)$43

the bilinear form

$X=F^{-1}(\mathcal K)$44

is well defined and independent of local extensions of $X=F^{-1}(\mathcal K)$45. The second-order conditions are:

• Necessary under LICQ:

$X=F^{-1}(\mathcal K)$46

• Sufficient under ZKRCQ plus strictness:

$X=F^{-1}(\mathcal K)$47

The paper emphasizes invariance: the tangent cone $X=F^{-1}(\mathcal K)$48, dual cone $X=F^{-1}(\mathcal K)$49, Lagrange multipliers $X=F^{-1}(\mathcal K)$50, and the Hessian $X=F^{-1}(\mathcal K)$51 are invariant under change of adapted local chart on $X=F^{-1}(\mathcal K)$52 and under equivalent reformulations $X=F^{-1}(\mathcal K)$53 where $X=F^{-1}(\mathcal K)$54 preserves $X=F^{-1}(\mathcal K)$55 (Bergmann et al., 2021).

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

A log differentiable space is a locally ringed space $X=F^{-1}(\mathcal K)$56 over $X=F^{-1}(\mathcal K)$57 equipped with a log structure

$X=F^{-1}(\mathcal K)$58

where $X=F^{-1}(\mathcal K)$59 is a sheaf of commutative monoids and $X=F^{-1}(\mathcal K)$60 induces an isomorphism

$X=F^{-1}(\mathcal K)$61

Equivalently, one works with the submonoid $X=F^{-1}(\mathcal K)$62 of non-negative functions and a map

$X=F^{-1}(\mathcal K)$63

In this language, a manifold with corners is precisely a positive log differentiable space which is log smooth and locally modeled on $X=F^{-1}(\mathcal K)$64 (Gillam et al., 2015).

On $X=F^{-1}(\mathcal K)$65, the chart monoid homomorphism is

$X=F^{-1}(\mathcal K)$66

where $X=F^{-1}(\mathcal K)$67 is the $X=F^{-1}(\mathcal K)$68-th coordinate function. After taking the associated log structure, one obtains a sheaf of monoids $X=F^{-1}(\mathcal K)$69 with a map $X=F^{-1}(\mathcal K)$70. The characteristic monoid at a point $X=F^{-1}(\mathcal K)$71 is

$X=F^{-1}(\mathcal K)$72

naturally isomorphic to $X=F^{-1}(\mathcal K)$73, and on stalks one has the exact sequence

$X=F^{-1}(\mathcal K)$74

which is split on manifolds with corners by taking the chart (Gillam et al., 2015).

The paper’s chart criterion for log smoothness states that a morphism $X=F^{-1}(\mathcal K)$75 of fine log differentiable spaces is log smooth if locally on $X=F^{-1}(\mathcal K)$76 there exist charts

$X=F^{-1}(\mathcal K)$77

together with a map of monoids $X=F^{-1}(\mathcal K)$78 making

$X=F^{-1}(\mathcal K)$79

a classically smooth map of differentiable spaces. Log smooth morphisms are stable under composition, base-change, and are local in $X=F^{-1}(\mathcal K)$80 and $X=F^{-1}(\mathcal K)$81. In particular, the projection

$X=F^{-1}(\mathcal K)$82

is log smooth (Gillam et al., 2015).

The combinatorics of corners are encoded by fans. A fan is a sharp locally monoidal space locally isomorphic to $X=F^{-1}(\mathcal K)$83 for some fine monoid $X=F^{-1}(\mathcal K)$84. Its points are prime ideals, equivalently faces, of $X=F^{-1}(\mathcal K)$85, and the structure sheaf is the localization presheaf. In the corner-model case $X=F^{-1}(\mathcal K)$86, the faces are the coordinate hyperplanes. The “boundary” functor corresponds on fans to taking the disjoint union of the maximal proper faces (Gillam et al., 2015).

The resolution theory is formulated monoidally and then realized differentiably. For every fs monoid $X=F^{-1}(\mathcal K)$87 there is a canonical combinatorial sequence of blowups of faces

$X=F^{-1}(\mathcal K)$88

on $X=F^{-1}(\mathcal K)$89, functorial in $X=F^{-1}(\mathcal K)$90, so that the final blowup $X=F^{-1}(\mathcal K)$91 is free, equivalently $X=F^{-1}(\mathcal K)$92. Passing to log differentiable spaces via $X=F^{-1}(\mathcal K)$93, these blowups pull back to Euclidean-proper, locally-projective, log smooth maps

$X=F^{-1}(\mathcal K)$94

which over the smooth locus are isomorphisms. More generally, if $X=F^{-1}(\mathcal K)$95 is any fs log differentiable space, one applies the functorial blowup on the associated fan $X=F^{-1}(\mathcal K)$96 to obtain a free fan $X=F^{-1}(\mathcal K)$97; the terminal object in the Kato-category lifts this subdivision to a log smooth, Euclidean-proper, surjective map of PLDS

$X=F^{-1}(\mathcal K)$98

with $X=F^{-1}(\mathcal K)$99 free, that is, a genuine manifold with corners. Over the locus where $\mathcal K\subset \mathcal N$00 was already free, the blowup is an isomorphism (Gillam et al., 2015).

The examples in the paper include a linear inequality region in $\mathcal K\subset \mathcal N$01, such as $\mathcal K\subset \mathcal N$02, 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 $\mathcal K\subset \mathcal N$03, with corner $\mathcal K\subset \mathcal N$04, and the space of fields is

$\mathcal K\subset \mathcal N$05

equipped with a local symplectic form $\mathcal K\subset \mathcal N$06, acted on by a local gauge group $\mathcal K\subset \mathcal N$07 with Lie algebra $\mathcal K\subset \mathcal N$08. The local momentum form $\mathcal K\subset \mathcal N$09 is defined by

$\mathcal K\subset \mathcal N$10

with integrated map

$\mathcal K\subset \mathcal N$11

By the Takens-Zuckerman decomposition,

$\mathcal K\subset \mathcal N$12

with $\mathcal K\subset \mathcal N$13 of order $\mathcal K\subset \mathcal N$14 and exact term $\mathcal K\subset \mathcal N$15, so that

$\mathcal K\subset \mathcal N$16

Here $\mathcal K\subset \mathcal N$17 is equivariant for the action of the constraint gauge subgroup $\mathcal K\subset \mathcal N$18, while $\mathcal K\subset \mathcal N$19 is a momentum map, up to cocycle, for the residual action of $\mathcal K\subset \mathcal N$20 (Riello et al., 2022).

The constraint set is

$\mathcal K\subset \mathcal N$21

The paper proves that $\mathcal K\subset \mathcal N$22 is a coisotropic submanifold in $\mathcal K\subset \mathcal N$23:

$\mathcal K\subset \mathcal N$24

Equivalently, in the theorem stated in the details,

$\mathcal K\subset \mathcal N$25

The ideal $\mathcal K\subset \mathcal N$26, where $\mathcal K\subset \mathcal N$27, is the maximal ideal for which $\mathcal K\subset \mathcal N$28 factors through $\mathcal K\subset \mathcal N$29 and $\mathcal K\subset \mathcal N$30 (Riello et al., 2022).

Reduction proceeds by stages. The first stage is constraint reduction at $\mathcal K\subset \mathcal N$31 by $\mathcal K\subset \mathcal N$32:

$\mathcal K\subset \mathcal N$33

with

$\mathcal K\subset \mathcal N$34

Equivalently,

$\mathcal K\subset \mathcal N$35

The second stage is flux superselection by $\mathcal K\subset \mathcal N$36. For each coadjoint orbit $\mathcal K\subset \mathcal N$37, define

$\mathcal K\subset \mathcal N$38

Point reduction at $\mathcal K\subset \mathcal N$39 yields

$\mathcal K\subset \mathcal N$40

The disjoint union

$\mathcal K\subset \mathcal N$41

carries a natural Poisson structure whose symplectic leaves are exactly the $\mathcal K\subset \mathcal N$42 (Riello et al., 2022).

Corner data are encoded by an action Lie algebroid restricted to $\mathcal K\subset \mathcal N$43:

$\mathcal K\subset \mathcal N$44

The $\mathcal K\subset \mathcal N$45-form

$\mathcal K\subset \mathcal N$46

is weakly symplectic and basic for the projection to $\mathcal K\subset \mathcal N$47. Hence $\mathcal K\subset \mathcal N$48 becomes a symplectic Lie algebroid, and its anchor $\mathcal K\subset \mathcal N$49 induces on $\mathcal K\subset \mathcal N$50 a partial Poisson bivector

$\mathcal K\subset \mathcal N$51

Both the on-shell corner data $\mathcal K\subset \mathcal N$52 and the fully reduced $\mathcal K\subset \mathcal N$53 project onto the same base

$\mathcal K\subset \mathcal N$54

which parametrizes flux superselection sectors in both constructions (Riello et al., 2022).

For Yang-Mills on $\mathcal K\subset \mathcal N$55, with $\mathcal K\subset \mathcal N$56 compact or semisimple, $\mathcal K\subset \mathcal N$57, $\mathcal K\subset \mathcal N$58, $\mathcal K\subset \mathcal N$59, and

$\mathcal K\subset \mathcal N$60

the constraint form is $\mathcal K\subset \mathcal N$61, the flux form satisfies $\mathcal K\subset \mathcal N$62, and

$\mathcal K\subset \mathcal N$63

The constraint set is

$\mathcal K\subset \mathcal N$64

the first reduction is

$\mathcal K\subset \mathcal N$65

and the second reduction at orbit $\mathcal K\subset \mathcal N$66 yields

$\mathcal K\subset \mathcal N$67

Hence

$\mathcal K\subset \mathcal N$68

a Weinstein bundle (Riello et al., 2022).

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.