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Given any point $p$ on a surface $S$ of Gauss curvature -1, there exists an open neighborhood $U\subset S$ and $p$-centered coordinates $(x,y):U\to\mathbb{R}^2$, whose image is a domain $R = (x,y)(U)\ … I}$ above; this immersion is unique up to rigid motion and has Gauss curvature $-1$. The asymptotic curves of the immersion $X$ are given by holding $x+y$ or $x-y$ constant. …

Remark: To see another application of the formula (1), one might consult this answer of mine, where it is used to compute the explicit distance function for the complete metric of negative curvature on …

Well the standard techniques would take advantage of the fact that the equation doesn't explicitly involve the independent variable $x$ to integrate the equation once, thereby leading to the conservat …

$ with the property that $H$ is nonconstant while $H^2-K>0$ is constant (where 'Bonnet surface' means that there is a nontrivial isometric deformation of $S$ in $\mathbb{R}^3$ that preserves the mean curvature … combining it with the three equations that follow from requiring that the surface admit a nontrvial $1$-parameter family of isometric embeddings into $\mathbb{R}^3$, all with the same nonconstant mean curvature …

Most of Cartan's isoparametric hypersurfaces in $S^4$ have your desired properties: They have constant principal curvatures (in fact, they are homogeneous), nearly all of them have all three principa …

You can always do this, but it's not as simple as using some kind of ODE (such a flow of vector fields) to construct such charts. First, assume that your surface is connected and simply-connected. Th …

It should not be surprising that, for a $2$-dimensional manifold, the Gauss curvature $K:M\to\mathbb{R}$ does not determine a unique metric $g$. … This is why one has the level of rigidity for the sectional curvature that is indicated in Kulkarni's and Yau's results. …

Élie Cartan proposes such interpretations in his fundamental paper Sur les variétés à connexion affine et la theorie de la relativité généralisée (Ann. Ec. Norm. 40 (1923), 325–412 and 41 (1924), 1–25 …

After all, in the example I mention in my comment above, that of the $3$-sphere with $g_1$ being the standard metric with constant sectional curvature $1$ and $g_2=\lambda g_1$ for some constant $\lambda … wedge\omega_{32} -\mu_3(\omega_{31}\wedge\omega_2 + \omega_1\wedge\omega_{32}) + (e^{2\mu}R^\ast_{1212}+{\mu_3}^2)\ \omega_1\wedge\omega_2\ . $$ The condition that the surface $S$ have the same Gauss curvature …

k_n} $$ where $C_{ijk_1\cdots k_n}$ is the value at $p$ of a tensor of the form $$ -\frac{2(n{-}1)}{n{+}1}\,\nabla_{k_3}\cdots\nabla_{k_n} R_{ik_1jk_2} + LOT_n $$ where $LOT_n$ is a polynomial in the curvature … Letting $r^2 = x^2 + y^2$ and letting $T$ be the radial vector field $x\,\partial_x + y\,\partial_y$, one computes the formula for the Gauss curvature of $g$ to be $$ K = -\frac{2(1+r^2h)(TTh) - r^2(Th …

It seems that the OP wants to be able to test whether a $(1,3)$-tensor $R$ with the symmetries of a Riemann curvature tensor is actually the curvature of a Riemannian metric. … For example, if $g_0$ is the standard metric on the $2$-sphere with Gauss curvature $+1$ and $R_0$ is its $(1,3)$-curvature tensor, then $-R_0$ cannot be the curvature tensor of any nondegenerate $g$ globally …

with the flat metric $dx^2+2dy^2$ outside of $D_2$ but has nonzero curvature somewhere inside $D_2$. … One can even arrange that the curvature of $g_i$ be nonzero in $D_i$ away from some closed subset $K_i$ that lies in the interior of $D_i$, so do this. …

Then, of course, $2F$ will satisfy this genericity condition as well, and, since it satisfies $d(2F) = (2F)\wedge A - A \wedge (2F)$, it follows that the only possible $1$-form whose curvature could be … However, the curvature of $A$ is $F\not=2F$. Thus, $2F$ satisfies your condition, but it is not a curvature form. …

The existence certainly remains true if the ambient metric is real-analytic, and this follows from the Cartan-Kähler Theorem since all you are asking for is local minimal surfaces. In the case of a …

If you just calculate using the moving frame, you'll get the answer for the variation of the principal curvatures in a few lines: $$ \delta\kappa_i = \mathrm{Hess}(u)(e_i,e_i) + \kappa_i^2\,u . $$ Her …