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Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise smooth loop $\gamma$, parallel transport any vector from $x \in \gamma$ back to $x$, the difference in angledifference in angle (holonomy) can be calculated by integrating the scalar curvaturethe scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting acting on the right). A connection of $\Pi$ is a    $G$-invariant    $\mathfrak{g}$-valued $1$-form on $P$ whose restriction restriction to each fiber fiber is the Maurer-Cartan $1$-form on $G$ torsor torsor. By Cartan structure structure theorem, the curvature $2$-form    $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$ $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly theexactly the integration of the curvature over any surface which cobounds cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2.   2. That the curvature vanishes is equivalent to the the connection is flat flat should also be weaker too.

After thoughts

  • From Roberto Ladu's answer, a key to understand this is the Non-abelian Stoke's theorem. An illustrative and beginner case (dimension=$1+1$) is clearly given in [2]

Reference

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise smooth loop $\gamma$, parallel transport any vector from $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a  $G$-invariant  $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form  $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2.   That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise smooth loop $\gamma$, parallel transport any vector from $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a  $G$-invariant  $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form  $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for. 2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

After thoughts

  • From Roberto Ladu's answer, a key to understand this is the Non-abelian Stoke's theorem. An illustrative and beginner case (dimension=$1+1$) is clearly given in [2]

Reference

deleted 1 character in body
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Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise linear smooth curveloop $\gamma$, parallel transport any vector from $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a $G$-invariant $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise linear smooth curve $\gamma$, parallel transport any vector from $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a $G$-invariant $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise smooth loop $\gamma$, parallel transport any vector from $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a $G$-invariant $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.
Edit as Ramiro mentioned.
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Student
  • 5.2k
  • 11
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Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary piecewise linearcontractible and piecewise linear smooth curve $\gamma$, parallel transport transport any vector from    $x \in \gamma$ back to $x$, the difference in angle angle (holonomy) can can be calculated by integrating the scalar curvature curvature within the enclosed enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsorstorsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a $G$-invariant $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integrationthe integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary piecewise linear smooth curve $\gamma$, parallel transport any vector from  $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors). A connection of $\Pi$ is a $G$-invariant $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.

Holonomy and curvature may seem to be slightly advanced topics in geometry. However, their origins are easily imaginable. Namely, picture the surface of earth $S$, and pick an arbitrary contractible and piecewise linear smooth curve $\gamma$, parallel transport any vector from  $x \in \gamma$ back to $x$, the difference in angle (holonomy) can be calculated by integrating the scalar curvature within the enclosed surface.

Several years after Gauss , things like the above were generalized into the setting of principal $G$-bundles. However, intuitive explanation became difficult to find. This post is looking for a formal yet intuitive statement like the above, and a proof for it.

Formal Setup

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P \xrightarrow{\Pi} M$ a smooth principal $G$-bundle (i.e. a bundle whose fibers are $G$-torsors; as usual the structure group is $G$ acting on the right). A connection of $\Pi$ is a $G$-invariant $\mathfrak{g}$-valued $1$-form on $P$ whose restriction to each fiber is the Maurer-Cartan $1$-form on $G$ torsor. By Cartan structure theorem, the curvature $2$-form $\Omega$ satisfies $$\Omega = d\omega + \frac{1}{2} [\omega, \omega].$$

A fact is that this setting vastly generalized the classical setting, where $M$ is a Riemannian manifold, $G$ is $O(dim(M))$, $\Pi$ is the underlying $G$-bundle for the tangent bundle, and the connection is the one associated to the Levi-Civita connection.

Wonder

I wonder if we can still make sense of the intuitive statement, that the holonomy along any contractible loop $\gamma$ is exactly the integration of the curvature over any surface which cobounds $\gamma$.

Naively, this does not make sense, as we need the curvature $2$-form to be on $M$, while the definition suggests that $\Omega$ is really a $2$-form on the bundle space $P$. There does not seem to be a natural $2$-form on $M$ too in this picture.

Q. How to make sense of it? And how to prove it?

Remarks

  1. The Ambrose-Singer theorem should be weaker than what I'm hoping for.
  2. That the curvature vanishes is equivalent to the connection is flat should also be weaker too.

Reference

  • Foundations of Differential Geometry I [Kobayashi and Nomizu], Chapter 2.
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