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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.
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.
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