The first fundamental form is an intrinsic quantity Explain

The first fundamental form is an intrinsic quantity. Explain what this means. What does it measure? Give an example of another quantity that is intrinsic. The second fundamental form is an extrinsic quantity. Explain what, this means. What, is measured by the second fundamental form? Give an example of another quantity that is extrinsic.

Solution

The first fundamental form tells you how to compute the distances along the paths within the surface (it is just a Riemannian metric of the surface thought as a standalone manifold, that is if we forget about the embedding / immersion). This explains why it is also called the intrinsic metric.All coefficients of FFF, their partial derivatives w.r.t. u and v and and any combination thereof. E.g., Gauss curvature K, geodesic curvature, geodesic torsion, length, area, integral curvature, Christoffel symbols,isometry, Levi-Civita connection, tangential rotation, normal rotation in between principal planes etc. These are natural/intrinsic creatures of embedding

The second fundamental form describes how \"curved\" the embedding is, in other words, how the surface is located in the ambient space. It is a kind of derivative of the unit normal along the surface or, equivalently, the rate of change of the tangent planes, taken in various directions within the surface. Alternatively, it is called the shape tensor (it has a close relation to the shape, or Weingarten, operator), and is an extrinsic quantity in the sense that it depends on the embedding. The second fundamental form (SFF) cannot be measured without an immersion ... a field along the surface normal is a must. Normal curvature changes in isometry. coefficients of L,N and M change but not its determinant (LNM2).