I'm asking this question because a course change on differential geometry at my university has updated the wording from extrinsic manifold to intrinsic manifold.
This got me wonder as to what the difference between the two terms might be.
Can someone illuminate with an example of what would be considered an intrinsic and what would be considered extrinsic manifold. Just for clarification purposes.
Thanks
The two-sphere, $S^2$, can be thought of both intrinsically and extrinsically. From the extrinsic point of view, $S^2$ is inside of three-space defined by the equation $x^2+y^2+z^2=1$. Inside of $\mathbb{R}^3$, the two sphere is easy to understand using regular methods from calculus and all your intuition makes sense.
The intrinsic view of $S^2$ is done with charts, pieces glued together to make the sphere. Each piece is an open set in $R^2$. It is the maps that glue the pieces together that define the manifold. From this point of view, your intuition of tangent vector, normal vector, etc. can not be used. You only have the maps in hand and thus lose the structure of the manifold sitting in $\mathbb{R}^3$. There are advantages to the intrinsic view, but not in the sense of intuition.