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I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but it is more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at its impressive array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but I don't think it is much more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at its impressive array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but it is more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at its impressing array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but it is more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at its impressive array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but it is more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at it impressing array of applications to other geometries and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.

I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.

1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes. Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.

Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.

2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but it is more geometric than

Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.

The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.

3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".

To see the "geometry" in symplectic geometry, you also have to look at its impressing array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.