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add some commentary on the basepoint issue, and an appropriate tag.

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edited Mar 11, 2020 at 20:17

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There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you compute the holonomy around the boundary of the parallelogram $\{p + t_1 v + t_2 w : 0 \leq t_1 \leq a, 0 \leq t_2 \leq b\}$. The second-order Taylor expansion of this holonomy, as a function of $a$ and $b$ (centred at $a=b=0$) is given by:

$$Id_{T_p U} + ab R_p(v,w)$$

If you think carefully about this, what it tells you is that if you have any map $D^2 \to N$ where $N$ is a Riemann manifold, you can compute the holonomy around the boundary of the disc by an appropriate "integral" on the interior of the disc, of the Riemann curvature tensor. It's not an integral in the traditional sense, as you are performing a limit of a system of composites of functions. But in spirit, it is reasonably-close to an integral.

Other than being a limit of a composite of a large number of functions, a technical issue is the choice of basepoint and ensuring you are using an appropriate transport of the Riemann tensor back to one tangent space.

I imagine this observation has been written up somewhere in the literature. Who has proven this theorem, and where might it appear? Does this type of groupoid-y integration have a standard name?

There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you compute the holonomy around the boundary of the parallelogram $\{p + t_1 v + t_2 w : 0 \leq t_1 \leq a, 0 \leq t_2 \leq b\}$. The second-order Taylor expansion of this holonomy, as a function of $a$ and $b$ (centred at $a=b=0$) is given by:

$$Id_{T_p U} + ab R_p(v,w)$$

If you think carefully about this, what it tells you is that if you have any map $D^2 \to N$ where $N$ is a Riemann manifold, you can compute the holonomy around the boundary of the disc by an appropriate "integral" on the interior of the disc, of the Riemann curvature tensor. It's not an integral in the traditional sense, as you are performing a limit of a system of composites of functions. But in spirit, it is reasonably-close to an integral.

I imagine this observation has been written up somewhere in the literature. Who has proven this theorem, and where might it appear? Does this type of groupoid-y integration have a standard name?

There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you compute the holonomy around the boundary of the parallelogram $\{p + t_1 v + t_2 w : 0 \leq t_1 \leq a, 0 \leq t_2 \leq b\}$. The second-order Taylor expansion of this holonomy, as a function of $a$ and $b$ (centred at $a=b=0$) is given by:

$$Id_{T_p U} + ab R_p(v,w)$$

If you think carefully about this, what it tells you is that if you have any map $D^2 \to N$ where $N$ is a Riemann manifold, you can compute the holonomy around the boundary of the disc by an appropriate "integral" on the interior of the disc, of the Riemann curvature tensor. It's not an integral in the traditional sense, as you are performing a limit of a system of composites of functions. But in spirit, it is reasonably-close to an integral.

Other than being a limit of a composite of a large number of functions, a technical issue is the choice of basepoint and ensuring you are using an appropriate transport of the Riemann tensor back to one tangent space.

I imagine this observation has been written up somewhere in the literature. Who has proven this theorem, and where might it appear? Does this type of groupoid-y integration have a standard name?

riemannian-geometry curvature groupoids

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asked Mar 11, 2020 at 9:07

44.4k
2
139
245

Integrating the Riemann curvature tensor over a singular 2-disc

There's a classic characterization of the Riemann curvature tensor. Say, take a Riemann metric on an open subset $U$ of $\mathbb R^n$. Given a point $p \in U$ and two vectors $v,w \in T_p U$ you compute the holonomy around the boundary of the parallelogram $\{p + t_1 v + t_2 w : 0 \leq t_1 \leq a, 0 \leq t_2 \leq b\}$. The second-order Taylor expansion of this holonomy, as a function of $a$ and $b$ (centred at $a=b=0$) is given by:

$$Id_{T_p U} + ab R_p(v,w)$$

If you think carefully about this, what it tells you is that if you have any map $D^2 \to N$ where $N$ is a Riemann manifold, you can compute the holonomy around the boundary of the disc by an appropriate "integral" on the interior of the disc, of the Riemann curvature tensor. It's not an integral in the traditional sense, as you are performing a limit of a system of composites of functions. But in spirit, it is reasonably-close to an integral.

I imagine this observation has been written up somewhere in the literature. Who has proven this theorem, and where might it appear? Does this type of groupoid-y integration have a standard name?

riemannian-geometry curvature