# Questions tagged [curvature]

Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)

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### Invariant description of the Weitzenböck curvature operator by Bourguignon

I recently came across the paper Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein by Jean Pierre Bourguignon. What he shows in §8 is that the ...
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### Relation between mean curvature and conformal metric

We'll consider $(N, g)$ a Riamannian Manifold and $\overline{g} = e^{2f}g$ a conformal metric. Let M be a hypersurface in N, $\overline{H}_M$ and $H_M$ the mean curvature of M with respect to the ...
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### Constant Gaussian curvature disks

This question has also been posted on MSE, but maybe here is the right place to post it. Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose ...
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### the curvature wave equation

I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...
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### Manifolds with boundary admitting no closed embedded minimal hypersurface

The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
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### Geometric interpretation of the Weyl tensor?

The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops. Question: Is there a similarly direct geometric ...
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### Wasserstein space with strictly non-positive sectional curvature

Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$. Does it ...
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### Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully": Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
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### mean curvature for codimension $>1$?

The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
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Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k &... 0answers 124 views ### Curvature without any reference to embedding or coordinates On a 2 sphere, which is topologically just a compact simply connected 2 dim space, one can edow geometric structure by requiring each two geodesics have equidistant intersections. Is this enough to ... 1answer 203 views ### Positive scalar curvature on the total space of a circle bundle Let$(\Sigma_\gamma,g)$be a closed and orientable Riemannian surface of genus$\gamma \geq 1$,$(M^3,\tilde{g})$be a closed, connected and orientable Riemannian$3$-manifold, and$\pi : M \to \...
Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L. In this later paper Chakerian gave a simpler proof and noted that ...
Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...