Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector field on $\partial M$. Is this normal field always the outward normal?

EDIT: It seems as if it depends on the dimension of the ambient space. For example, take $\mathbb{B}^3 \subset \mathbb{R}^3$ oriented naturally. Give $\partial \mathbb{B}^3 = \mathbb{S}^2$ the induced orientation. Then the normal vector field on $\mathbb{S}^2$ points outward. But now look at $\mathbb{B}^2 \subset \mathbb{R}^2$ oriented naturally. Its boundary is simply $\mathbb{S}^1$ and has an inward pointing normal vector field. I've spared you the details of my reasoning, but please tell me if my conclusions are correct? If I am correct, then don't we make the convention such that the induced orientation always results in an outward pointing normal vector?

EDIT 2: I made a mistake in my second example - the normal field is again outward pointing. So I guess it is always outward pointing?

  • $\begingroup$ "Give $\partial M$ the induced orientation." What did you intend this to mean? $\endgroup$ – PVAL-inactive Mar 31 '16 at 12:34
  • $\begingroup$ Inward and outward normal vectors have difference ? $\endgroup$ – HK Lee Mar 31 '16 at 12:49
  • $\begingroup$ There are several (two) conventions about orientation "induced". Also, outward has meaning in an ambience space and codimension $1$, otherwise you need to define it. $\endgroup$ – Z. L. Mar 31 '16 at 18:02

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