Difference between surface indices and ambient indices?

Solution 1:

Let us use an example of a 2D surface embedded in a 3D space and a vector field (an instance of a tensor field) defined on the embedded surface. In this scenario, the vector field often is assumed to lie in the tangent space of each point, so the component representation of the vector field requires two "surface components" that are denoted by "surface indices."

In that section of the book, however, the author investigates the case when the vector field also has a "surface normal component" outside of the tangent space. In this case, the component representation of the vector field needs to have three components, which are denoted by the "ambient indices."

If you are still unsure, it may be helpful to see the author's Youtube lecture video on this section: https://www.youtube.com/watch?v=8VGUooPINYg

I hope this information helps.