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There seems to be some confusion over what the tangent space to a singular point of an orbifold is.

On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-invariant structures on $\mathbb R^n$ ($G$ being the finite group so that the orbifold is locally (about some specific point $x$) the quotient of $\mathbb R^n$ by the action of $G$). One might be tempted to consider cone points as differentiable spaces (that is, subsets of some $\mathbb R^k$, inheriting their differential structure by restriction), however, we are told, for example, that $\mathbb R^2/\mathbb Z_3$ and $\mathbb R^2/\mathbb Z_4$ are distinct as orbifolds, so it is not the case that cone points can be modeled merely with cone-like subsets of some $\mathbb R^k$. The definition in which 'smooth' means 'lifts to $G$-invariant smooth' distinguishes these two cones, as the set of functions with 3-fold symmetry and the set of functions with 4-fold symmetry, in $\mathbb R^2$, are distinct. The third item in Satake's seminal paper [On a Generalization of the Notion of Manifold] corroborates this, giving $\mathbb C^\infty$ forms of degree $p$ at a singularity $x$ as those $\mathbb C^\infty$ $p$-forms in $\mathbb R^n$ which are invariant under $G_x$. If we require the same property of vectors, that is, that they lift to $G$-invariant vectors in $\mathbb R^n$, then we have that the dimension of the tangent space of an orbifold is the dimension of the invariant subspace upstairs. In particular the dimension tends to drop at the singular points. For example, the dimension of the tangent space at the singularity in $\mathbb R^2/\mathbb Z_3$ is 0. This notion of vector agrees with the notion of vector as derivation on the germ of smooth functions. In this case smooth functions in $\mathbb R^2$ which have 3-fold symmetry necessarily have vanishing derivatives at the origin.

On the other hand, one finds descriptions of smooth orbifolds as objects which have tangent bundle-like structures, which are locally $\mathbb R^n/G$. It is not clear what this means as far as smooth structures go, but the explanation above of $\mathbb R^2/\mathbb Z_3$ having a 0 dimensional tangent space at the cone point seems to contradict the notion that the tangent-like space at the singularity in $\mathbb R^2/\mathbb Z_3$ is $\mathbb R^2/\mathbb Z_3$, whatever that means.

It is also said that manifolds with boundary can be viewed as orbifolds, which have isotropy group reflection by $\mathbb Z_2$ along their boundaries. It would be nice to include the note that the differentiable structures are different. Specifically, smooth manifolds with boundaries have tangent spaces along their boundaries which are the same dimension as the manifold. In contrast, the same topological space as an orbifold with $\mathbb Z_2$ structure group along the boundary should have a tangent space which is one dimension less than the dimension at a generic point, if the definition of tangent space follows Satake's guideline. Indeed, smooth functions in $\mathbb R^n$ which locally have symmetry by reflection through a codimension 1 hyperplane, have vanishing partial derivatives in the normal direction.

I am asking for concurrence or correction and clarification, since I am still not certain I have the correct notion of tangent space to an orbifold, although I'm fairly confident in the first given here.

2 improved sentence

There seems to be some confusion over what the tangent space to a singular point of an orbifold is.

On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-invariant structures on $\mathbb R^n$ ($G$ being the finite group so that the orbifold is locally (about some specific point $x$) the quotient of $\mathbb R^n$ by the action of $G$). One might be tempted to consider cone points as differentiable spaces (that is, subsets of some $\mathbb R^k$, inheriting their differential structure by restriction), however, we are told, for example, that $\mathbb R^2/\mathbb Z_3$ and $\mathbb R^2/\mathbb Z_4$ are distinct as orbifolds, so it is not the case that cone points can be modeled merely with cone-like subsets of some $\mathbb R^k$. The definition in which 'smooth' means 'lifts to $G$-invariant smooth' distinguishes these two cones, as the set of functions with 3-fold symmetry and the set of functions with 4-fold symmetry, in $\mathbb R^2$, are distinct. The third item in Satake's seminal paper [On a Generalization of the Notion of Manifold] corroborates this, giving $\mathbb C^\infty$ forms of degree $p$ at a singularity $x$ as those $\mathbb C^\infty$ $p$-forms in $\mathbb R^n$ which are invariant under $G_x$. If we require the same property of vectors, that is, that they lift to $G$-invariant vectors in $\mathbb R^n$, then we have that the dimension of the tangent space of an orbifold is the dimension of the invariant subspace upstairs. In particular the dimension tends to drop at the singular points. For example, the dimension of the tangent space at the singularity in $\mathbb R^2/\mathbb Z_3$ is 0. This notion of vector agrees with the notion of vector as derivation on the germ of smooth functions. In this case smooth functions in $\mathbb R^2$ which have 3-fold symmetry necessarily have vanishing derivatives at the origin.

On the other hand, one finds descriptions of smooth orbifolds as objects which have tangent bundle-like structures, which are locally $\mathbb R^n/G$. It is not clear what this means as far as smooth structures go, but the explanation above of $\mathbb R^2/\mathbb Z_3$ having a 0 dimensional tangent space at the cone point seems to contradict the notion that the tangent-like space at the singularity in $\mathbb R^2/\mathbb Z_3$ is $\mathbb R^2/\mathbb Z_3$, whatever that means.

It is also said that manifolds with boundary can be viewed as orbifolds, which have isotropy group reflection by $\mathbb Z_2$ along their boundaries. It would be nice to include the note that the differentiable structures are different. Specifically, smooth manifolds with boundaries have tangent spaces along their boundaries which are the same dimension as the manifold. In contrast, the same topological space as an orbifold with $\mathbb Z_2$ structure group along the boundary should have a tangent space which is one dimension less than the dimension at a generic point, if the definition of tangent space follows Satake's guideline. Indeed, smooth functions in $\mathbb R^n$ which locally have symmetry by reflection through a codimension 1 hyperplane, have vanishing partial derivatives in the normal direction.

I am asking for concurrence or correction and clarification, since I am still not certain I have the correct notion of tangent space to an orbifold, although I'm fairly confident in the first given here.

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