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Results tagged with curvature
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user 129995
Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
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Gaussian curvature and curvature of the Levi-Civita connection
The curvature of the connection is a matrix of 2-forms $\Omega=d\Theta$. … .
$$
I know that the coefficient $-\left(e_1(T_{12}^2)-e_2(T_{12}^1)+(T_{12}^1)^2+(T_{12}^2)^2\right)$ is indeed the Gaussian curvature of the surface. But I am not able to prove it. …
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Gaussian curvature and curvature of the Levi-Civita connection
We are going to show it is indeed the Gaussian curvature in case $S$ is an immersed surface $\iota:S\to \mathbb{R}^3$. … Now, we only have to put together several things:
First, we know that Gaussian curvature $K$, as the product of the main curvatures, is the determinant of the shape operator. …