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Symplectic Geometry

This quiz covers the fundamental concepts and applications of Symplectic Geometry, a branch of differential geometry that studies symplectic manifolds and their properties.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a symplectic manifold?

  1. A manifold equipped with a symplectic form.
  2. A manifold with a Riemannian metric.
  3. A manifold with a complex structure.
  4. A manifold with a flat connection.
Question 2 Multiple Choice (Single Answer)

What is a symplectic form?

  1. A closed, non-degenerate 2-form.
  2. A closed, non-degenerate 1-form.
  3. An exact 2-form.
  4. A harmonic 2-form.
Question 3 Multiple Choice (Single Answer)

What is the symplectic gradient of a function?

  1. The vector field \(X_f\) defined by \(\omega(X_f, Y) = df(Y)\) for all tangent vectors \(Y\).
  2. The vector field \(X_f\) defined by \(\omega(X_f, Y) = -df(Y)\) for all tangent vectors \(Y\).
  3. The vector field \(X_f\) defined by \(\omega(X_f, Y) = f(Y)\) for all tangent vectors \(Y\).
  4. The vector field \(X_f\) defined by \(\omega(X_f, Y) = -f(Y)\) for all tangent vectors \(Y\).
Question 4 Multiple Choice (Single Answer)

What is the Hamiltonian vector field of a function?

  1. The vector field \(X_H\) defined by \(\omega(X_H, Y) = dH(Y)\) for all tangent vectors \(Y\).
  2. The vector field \(X_H\) defined by \(\omega(X_H, Y) = -dH(Y)\) for all tangent vectors \(Y\).
  3. The vector field \(X_H\) defined by \(\omega(X_H, Y) = H(Y)\) for all tangent vectors \(Y\).
  4. The vector field \(X_H\) defined by \(\omega(X_H, Y) = -H(Y)\) for all tangent vectors \(Y\).
Question 5 Multiple Choice (Single Answer)

What is the Poisson bracket of two functions?

  1. The function \(\{f, g\}\) defined by \(\{f, g\}(x) = \omega(X_f(x), X_g(x))\).
  2. The function \(\{f, g\}\) defined by \(\{f, g\}(x) = -\omega(X_f(x), X_g(x))\).
  3. The function \(\{f, g\}\) defined by \(\{f, g\}(x) = f(x)g(x)\).
  4. The function \(\{f, g\}\) defined by \(\{f, g\}(x) = -f(x)g(x)\).
Question 6 Multiple Choice (Single Answer)

What is the symplectic form on the cotangent bundle of a manifold?

  1. The canonical symplectic form \(\omega = d\theta\) on the cotangent bundle \(T^*M\).
  2. The canonical symplectic form \(\omega = -d\theta\) on the cotangent bundle \(T^*M\).
  3. The canonical symplectic form \(\omega = \theta\) on the cotangent bundle \(T^*M\).
  4. The canonical symplectic form \(\omega = -\theta\) on the cotangent bundle \(T^*M\).
Question 7 Multiple Choice (Single Answer)

What is the symplectic form on the phase space of a Hamiltonian system?

  1. The canonical symplectic form \(\omega = d\theta\) on the phase space \(\mathbb{R}^{2n}\).
  2. The canonical symplectic form \(\omega = -d\theta\) on the phase space \(\mathbb{R}^{2n}\).
  3. The canonical symplectic form \(\omega = \theta\) on the phase space \(\mathbb{R}^{2n}\).
  4. The canonical symplectic form \(\omega = -\theta\) on the phase space \(\mathbb{R}^{2n}\).
Question 8 Multiple Choice (Single Answer)

What is the symplectic form on the space of loops in a symplectic manifold?

  1. The Weil-Petersson symplectic form \(\omega = \int_\gamma \omega\) on the space of loops in a symplectic manifold \(M\).
  2. The Weil-Petersson symplectic form \(\omega = -\int_\gamma \omega\) on the space of loops in a symplectic manifold \(M\).
  3. The Weil-Petersson symplectic form \(\omega = \int_\gamma \theta\) on the space of loops in a symplectic manifold \(M\).
  4. The Weil-Petersson symplectic form \(\omega = -\int_\gamma \theta\) on the space of loops in a symplectic manifold \(M\).
Question 9 Multiple Choice (Single Answer)

What is the symplectic form on the space of paths in a symplectic manifold?

  1. The Hofer-Zehnder symplectic form \(\omega = \int_\gamma \omega\) on the space of paths in a symplectic manifold \(M\).
  2. The Hofer-Zehnder symplectic form \(\omega = -\int_\gamma \omega\) on the space of paths in a symplectic manifold \(M\).
  3. The Hofer-Zehnder symplectic form \(\omega = \int_\gamma \theta\) on the space of paths in a symplectic manifold \(M\).
  4. The Hofer-Zehnder symplectic form \(\omega = -\int_\gamma \theta\) on the space of paths in a symplectic manifold \(M\).
Question 10 Multiple Choice (Single Answer)

What is the symplectic form on the space of Hamiltonian diffeomorphisms of a symplectic manifold?

  1. The Arnold-Liouville symplectic form \(\omega = \int_M \omega\) on the space of Hamiltonian diffeomorphisms of a symplectic manifold \(M\).
  2. The Arnold-Liouville symplectic form \(\omega = -\int_M \omega\) on the space of Hamiltonian diffeomorphisms of a symplectic manifold \(M\).
  3. The Arnold-Liouville symplectic form \(\omega = \int_M \theta\) on the space of Hamiltonian diffeomorphisms of a symplectic manifold \(M\).
  4. The Arnold-Liouville symplectic form \(\omega = -\int_M \theta\) on the space of Hamiltonian diffeomorphisms of a symplectic manifold \(M\).
Question 11 Multiple Choice (Single Answer)

What is the symplectic form on the space of Lagrangian submanifolds of a symplectic manifold?

  1. The Weinstein symplectic form \(\omega = \int_L \omega\) on the space of Lagrangian submanifolds of a symplectic manifold \(M\).
  2. The Weinstein symplectic form \(\omega = -\int_L \omega\) on the space of Lagrangian submanifolds of a symplectic manifold \(M\).
  3. The Weinstein symplectic form \(\omega = \int_L \theta\) on the space of Lagrangian submanifolds of a symplectic manifold \(M\).
  4. The Weinstein symplectic form \(\omega = -\int_L \theta\) on the space of Lagrangian submanifolds of a symplectic manifold \(M\).
Question 12 Multiple Choice (Single Answer)

What is the symplectic form on the space of symplectic embeddings of a symplectic manifold into another symplectic manifold?

  1. The Gromov symplectic form \(\omega = \int_M \omega_1 - \int_N \omega_2\) on the space of symplectic embeddings of a symplectic manifold \(M\) into another symplectic manifold \(N\).
  2. The Gromov symplectic form \(\omega = -\int_M \omega_1 + \int_N \omega_2\) on the space of symplectic embeddings of a symplectic manifold \(M\) into another symplectic manifold \(N\).
  3. The Gromov symplectic form \(\omega = \int_M \theta_1 - \int_N \theta_2\) on the space of symplectic embeddings of a symplectic manifold \(M\) into another symplectic manifold \(N\).
  4. The Gromov symplectic form \(\omega = -\int_M \theta_1 + \int_N \theta_2\) on the space of symplectic embeddings of a symplectic manifold \(M\) into another symplectic manifold \(N\).
Question 13 Multiple Choice (Single Answer)

What is the symplectic form on the space of Hamiltonian actions of a Lie group on a symplectic manifold?

  1. The Marsden-Weinstein symplectic form \(\omega = \int_M \omega - \int_G \theta\) on the space of Hamiltonian actions of a Lie group \(G\) on a symplectic manifold \(M\).
  2. The Marsden-Weinstein symplectic form \(\omega = -\int_M \omega + \int_G \theta\) on the space of Hamiltonian actions of a Lie group \(G\) on a symplectic manifold \(M\).
  3. The Marsden-Weinstein symplectic form \(\omega = \int_M \theta - \int_G \omega\) on the space of Hamiltonian actions of a Lie group \(G\) on a symplectic manifold \(M\).
  4. The Marsden-Weinstein symplectic form \(\omega = -\int_M \theta + \int_G \omega\) on the space of Hamiltonian actions of a Lie group \(G\) on a symplectic manifold \(M\).
Question 14 Multiple Choice (Single Answer)

What is the symplectic form on the space of symplectic vector spaces?

  1. The Kostant-Kirillov symplectic form \(\omega = \int_V \omega\) on the space of symplectic vector spaces \(V\).
  2. The Kostant-Kirillov symplectic form \(\omega = -\int_V \omega\) on the space of symplectic vector spaces \(V\).
  3. The Kostant-Kirillov symplectic form \(\omega = \int_V \theta\) on the space of symplectic vector spaces \(V\).
  4. The Kostant-Kirillov symplectic form \(\omega = -\int_V \theta\) on the space of symplectic vector spaces \(V\).
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