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Post Closed as "Not suitable for this site" by Will Jagy, Ryan Budney, abx, Denis Serre, Stefan Kohl♦
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user62013
user62013
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Let $M$$M \subset \mathbb{R}^d$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) \geq 90^{\circ}$? It seems intuitively obvious but how does one formally prove this?

Let $M$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) \geq 90^{\circ}$? It seems intuitively obvious but how does one formally prove this?

Let $M \subset \mathbb{R}^d$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) \geq 90^{\circ}$? It seems intuitively obvious but how does one formally prove this?

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user62013
user62013
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Normals along a Sphere

Let $M$ be a smooth 2-manifold that is homeomorphic to a sphere or a connected sum or tori. Does there always exists two points $x,y \in M$ such that the normals $\angle(n_x, n_y) \geq 90^{\circ}$? It seems intuitively obvious but how does one formally prove this?