Tagged Questions

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Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
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How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3$$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
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Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
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The Definition of the Second Fundamental Form

Let $r:M\rightarrow{\mathbb{R}^{n+1}}$ be an isometric immersion and $M$ is an $n$-dimensional Riemannian Manifold. That is to say, $M$ is the hypersurface in $\mathbb{{R}^{n+1}}$. Then we can ...
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Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
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Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
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Formula for curvature of two intersecting surfaces in terms of their normal curvature.

I have been privately reading DoCarmo recently, and have been attempting to do some of the problems. I am stuck on this one, it is problem 14 in section 3.2 for those interested. If someone could show ...
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Why does it appear that Willmore energy is always zero?

The answer is "because I'm being sloppy," but the problem is I don't know exactly where I'm being sloppy. Here's my sloppy argument: Let $M$ be a smooth compact surface without boundary in ...
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Christoffel symbols in Differential geometry iff proof

I need help in proving that $H = 0$ for a surface iff $g_{11}L_{22} - 2g_{12}L_{12} + g_{22}L_{11} = 0.$ I think that these are the Christoffel symbols exploited in some manner and normally, I'm not ...
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great circle distance

in the euclidean plane the distance from the origin to a point is $s^2 = x^2 + y^2$ I am reading a paper which say that this could be called an algabraic metric for the plane. the paper then ...