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Results tagged with isometric-immersion
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user 46290
3
votes
0
answers
100
views
Is there a weak isometric completion to a $W^{2,2}$ isometric immersion?
Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$.
Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$).
Suppose we are given a $W^{2,2}$ isometric immersion $F: …
- 6,741
9
votes
2
answers
470
views
Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?
It is well known that there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$, and that this embedding cannot be $C^2$.
Is there a $W^{2,2}$ isometric embedding? (i.e an isometric …
- 6,741
0
votes
Are all symmetries of the Dirichlet functional isometries?
This is just an elaboration on Robert's great answer:
The key idea is to use the fact that the induced metric on the Hom -space of two inner product spaces is "linear" in the "metric" on the target.
…
6
votes
2
answers
205
views
Are all symmetries of the Dirichlet functional isometries?
This is a cross-post from MSE (no answer there).
Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth.
Consider the Dirichlet energy functional: $ …
- 6,741