Mean Curvature and Invariance.

[WWGD]

Hi All:
I am curious about the definition of mean curvature and its apparent lack of invariance under changes of coordinates: AFAIK, mean curvature is defined as the trace of the second fundamental form II(a,b). II(a,b) is a quadratic/bilinear form, and I do not see how its trace is invariant under (local, i.e., chartwise ) change of coordinate. I assume the solution to this (apparent) problem has to see with some result in multilinear algebra; specifically with the result that $Hom(W,W) ≈ W\otimes W^*$ , using the fact that every map $L: W \rightarrow W$ gives rise to a bilinear form ( assuming the presence of an inner-product ). Maybe we can go in the opposite direction and get the map L from the quadratic ( second fundamental) form and then compute its trace? Even if this is possible, can we guarantee that the trace of this map is independent of the (local/coordinate-wise) choice of coordinates?

P.S: Is there a way within the settings/menu of saving the work while posting? I just wasted around 20 minutes posting only to be told that I had to log in again to be able to post. I hit the back button in my computer, but all my work had been erased.

 

[jvicens]

Hi there,

Thank you for your question about the definition of mean curvature and its apparent lack of invariance under changes of coordinates. You are correct in stating that mean curvature is defined as the trace of the second fundamental form, which is a quadratic or bilinear form. This means that it is dependent on the choice of coordinates, as the values of the second fundamental form will change with different coordinate systems.

However, this does not mean that mean curvature is not a well-defined quantity. In fact, the trace of the second fundamental form can be computed using the map L that you mentioned, which is obtained from the quadratic form. This map is independent of the choice of coordinates and is therefore invariant under changes of coordinates.

To understand this better, we can look at the concept of tensor fields. The second fundamental form is a tensor field, which means it has different components in different coordinate systems. However, the trace of the second fundamental form is a scalar field, which means it remains the same regardless of the coordinate system. This is because the map L, which is used to compute the trace, is a tensor field and therefore its value remains the same in all coordinate systems.

In terms of saving your work while posting, most forums do not have a feature for this. It is always a good idea to save your work in a separate document before posting, just in case. I apologize for the inconvenience you experienced.

I hope this helps clarify the concept of mean curvature and its invariance under changes of coordinates. Let me know if you have any further questions.