grandes-ecoles 2022 Q48

grandes-ecoles · France · centrale-maths1__mp Groups Symplectic and Orthogonal Group Properties

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$, where $B^{2m}(r)$ is the closed Euclidean ball and $Z^{2m}(r)$ is the symplectic cylinder. Using the result of Q47, show that $1 \leqslant r$.

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$, where $B^{2m}(r)$ is the closed Euclidean ball and $Z^{2m}(r)$ is the symplectic cylinder. Using the result of Q47, show that $1 \leqslant r$.

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