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edited May 31, 2021 at 22:30

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As pointed out in the comments, the term geometry can have many meanings. Given a symplectic manifold $(M,\omega)$, I think almost (you never know!) all of us will agree that a Riemannian metric $g$ on $M$ would qualify as a geometric structure. While the symplectic form $\omega$ on its own does not provide a metric, the notion of an $\omega$-compatible or $\omega$-tame almost-complex structure on $M$ serves as a link between the symplectic structure and interesting Riemannian structures. See, for example, this MO questionAlmost Complex Structures: 'Tame' versus 'Compatible'.
Perhaps one can say that the symplectic structure of $M$ $-$— just like its topology $-$— imposes restrictions on (reasonable/interesting/compatible) Riemannian metrics available on $M$, and can in this sense be regarded as encoding certain rough geometric features shared by all (reasonable/interesting/compatible) geometries available on $M$. Whether this makes the symplectic structure itself a geometric structure now depends on the roughness level of one's personal definition of geometry.

As pointed out in the comments, the term geometry can have many meanings. Given a symplectic manifold $(M,\omega)$, I think almost (you never know!) all of us will agree that a Riemannian metric $g$ on $M$ would qualify as a geometric structure. While the symplectic form $\omega$ on its own does not provide a metric, the notion of an $\omega$-compatible or $\omega$-tame almost-complex structure on $M$ serves as a link between the symplectic structure and interesting Riemannian structures. See, for example, this MO question.
Perhaps one can say that the symplectic structure of $M$ $-$ just like its topology $-$ imposes restrictions on (reasonable/interesting/compatible) Riemannian metrics available on $M$, and can in this sense be regarded as encoding certain rough geometric features shared by all (reasonable/interesting/compatible) geometries available on $M$. Whether this makes the symplectic structure itself a geometric structure now depends on the roughness level of one's personal definition of geometry.

As pointed out in the comments, the term geometry can have many meanings. Given a symplectic manifold $(M,\omega)$, I think almost (you never know!) all of us will agree that a Riemannian metric $g$ on $M$ would qualify as a geometric structure. While the symplectic form $\omega$ on its own does not provide a metric, the notion of an $\omega$-compatible or $\omega$-tame almost-complex structure on $M$ serves as a link between the symplectic structure and interesting Riemannian structures. See, for example, Almost Complex Structures: 'Tame' versus 'Compatible'.
Perhaps one can say that the symplectic structure of $M$ — just like its topology — imposes restrictions on (reasonable/interesting/compatible) Riemannian metrics available on $M$, and can in this sense be regarded as encoding certain rough geometric features shared by all (reasonable/interesting/compatible) geometries available on $M$. Whether this makes the symplectic structure itself a geometric structure now depends on the roughness level of one's personal definition of geometry.

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created May 31, 2021 at 22:22

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As pointed out in the comments, the term geometry can have many meanings. Given a symplectic manifold $(M,\omega)$, I think almost (you never know!) all of us will agree that a Riemannian metric $g$ on $M$ would qualify as a geometric structure. While the symplectic form $\omega$ on its own does not provide a metric, the notion of an $\omega$-compatible or $\omega$-tame almost-complex structure on $M$ serves as a link between the symplectic structure and interesting Riemannian structures. See, for example, this MO question.
Perhaps one can say that the symplectic structure of $M$ $-$ just like its topology $-$ imposes restrictions on (reasonable/interesting/compatible) Riemannian metrics available on $M$, and can in this sense be regarded as encoding certain rough geometric features shared by all (reasonable/interesting/compatible) geometries available on $M$. Whether this makes the symplectic structure itself a geometric structure now depends on the roughness level of one's personal definition of geometry.