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specified spherical topology
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Josh Kirklin
Josh Kirklin
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Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?

I'm particularly interested in the case where $S$ has spherical topology.

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?

I'm particularly interested in the case where $S$ has spherical topology.

removed deprecated (geometry) tag - see the tag info: http://mathoverflow.net/tags/geometry/info; if there are some other geometry-related tags which are suitable, please use some of them instead
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Martin Sleziak
Martin Sleziak
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Integral of gaussianGaussian curvature multiplied by mean curvature

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-BonnetGauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?

Integral of gaussian curvature multiplied by mean curvature

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?

Integral of Gaussian curvature multiplied by mean curvature

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?

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Josh Kirklin
Josh Kirklin
  • 245
  • 1
  • 7

Integral of gaussian curvature multiplied by mean curvature

Let $M$ be a 3-manifold with positive definite metric $g$, and let $S\subset M$ be an oriented 2-surface. For $x\in S$ let $K(x)$ be the Gaussian curvature and $H(x)$ be the mean curvature of $S$ at the point $x$. Gauss-Bonnet tells us exactly what $\int_S K\,\mathrm{d}S$ is $-$ it is the Euler characteristic of $S$ (up to proportionality). Can a similarly simply interpretation be put to $\int_S KH\,\mathrm{d}S$?