3.7. SMOOTH MANIFOLD 21

Figure 3.7 represents the transition functions between two charts deﬁned on the same man-

ifold.

Figure 3.7: Given two neighborhoods U

i

and U

j

, each one with an associated coordinate map '

i

and

'

j

respectively, the transition function '

i;j

is deﬁned on the intersection of U

i

and U

j

; in particular

'

i;j

must be compliant with both the '

i

and '

j

[217].

3.7 SMOOTH MANIFOLD

Smooth manifolds are manifolds whose transition functions have derivatives of all orders. Smooth

surfaces, which are particularly important for shape analysis, belong to this class. Formally:

Smooth manifold A k-dimensional topological manifold with (resp. without) boundary is called

a smooth manifold or C

1

with (resp. without) boundary, if all transition functions '

i;j

are smooth.

If all transition functions '

i;j

are n times continuously diﬀerentiable, then the manifold is called

a C

n

manifold.

Smooth surface A smooth 2-dimensional manifold with (resp. without) boundary is called a

smooth surface, or simply surface, with (resp. without) boundary.

3.8 ORIENTABILITY

Orientability can be introduced in many ways. e intuition behind it is to be able to orient the

normals at every point of our object in a consistent manner, for instance by using the well-known

right-hand rule. Beside the standard Euclidean space, the concept of orientability can be extended

to more general spaces, such as manifolds, where the role of normals is taken by the Jacobian.²

A manifold M is called orientable if there exists an atlas f.U

i

; '

i

/g on it such that the

Jacobian of all transition functions '

i;j

from a chart to another is positive for all intersecting pairs

of regions. Manifolds that do not satisfy this property are called non-orientable.

²e Jacobian matrix extends the notion of gradient to multi-variate functions and represents the matrix of all the ﬁrst-partial

derivatives; the Jacobian is the determinant of the Jacobian matrix. For the notions of diﬀerentiability we refer the reader to

standard books of real analysis [118, 123].

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