Chapter 14

Integration on ManifoldsTill now, integrals have mostly pertained to measurable subsets of Rp and not somethinglike a surface contained in a higher dimensional space. This is what is considered in thischapter. First is an abstract description of manifolds and then an interesting application ofthe representation theorem for positive linear functionals is used to give a measure on amanifold. This is the higher dimensional version of arc length for a smooth curve seen incalculus.

Definition 14.0.1 Let S be a nonempty set in a metric space (X ,d). ∂S is the setof points x, if any with the property that B(x,r) contains points of S and points of X \S foreach r > 0. The interior of S consists of the union of all open subsets of S.

Lemma 14.0.2 Let U be a nonempty open set in a metric space (X ,d) . ∂U = Ū \U.

Proof: If x ∈ ∂U, then x can’t be in U because some ball containing x is contained inU . However, it must be in Ū because if not, some ball containing x would contain no pointsof Ū since Ū is closed.

If x ∈ Ū \U then if some ball containing x fails to contain other points which are in Uthen that ball would show x /∈ Ū . Hence every ball containing x must contain points of U .However, x itself is not in U and so x ∈ ∂U . ■

14.1 ManifoldsDefinition 14.1.1 An essential part of the definition of a manifold is the idea ofa relatively open set defined next. Recall that a homeomorphism is a one to one, onto,continuous mapping from one metric space to another which has continuous inverse. Ahalf space will be of the form {x : xi ≥ ai} or {x : xi ≤ ai} .

Definition 14.1.2 Let X be a metric space and let Ω⊆ X. Then a set U is called arelatively open set or open in Ω if it is the intersection of an open set of X with Ω. Thus Ω isa metric space with respect to the distance d (x,y) inherited from X and all considerationssuch as limit points, closures, limits, closed sets, open sets etc. in this metric space are takenwith respect to this metric. Continuity is also defined in terms of this metric on Ω inheritedfrom X. Ω is a p dimensional manifold with boundary if there is a locally finite cover {Ui}(here it will be a finite cover) of sets open in Ω such that each Ui is homeomorphic to a setopen in H where H is a half space or some finite intersection of such half spaces. Denotethe open sets and homeomorphisms by (Ui,Ri) . The collection of these is called an atlas.Thus RiUi is a set open in HRi where HRi is described above. Note that it could be a closedbox. Then a point x is called a boundary point if and only if Rix is a boundary point of theinterior of some HRi for some i.

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I will be assuming that Ω is compact and so we can replace “locally finite” with finitein the above definition. First I need to verify that the idea of ∂Ω is well defined.

Lemma 14.1.3 ∂Ω is well defined in the sense that the statement that x is a boundarypoint does not depend on which chart is considered.

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