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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1$\begingroup$ This looks like a DG homework assignment. $\endgroup$– Ryan BudneyCommented Nov 24, 2014 at 23:19
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1$\begingroup$ It's not, self study. $\endgroup$– user62013Commented Nov 24, 2014 at 23:55
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Assuming $M\subset \Bbb R^3$ and that is how you are measuring angles. Consider the map for each $x\in M$, $\phi_x: \Bbb R \rightarrow \Bbb R^3$ which moves along the normal at $x$. By the tranversality theorem, $\phi_x \pitchfork M$ for almost every $x\in M$. For all $x$, $|\phi_x(\Bbb R)\cap M| \geq 2$, so now you can just look at an intersection point adjacent to the one at $x$ (with no intersection points in between).
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$\begingroup$ Actually I intended for $M \subset \mathbb{R}^d$, codimension $\geq 1$, but I think pointing me toward the transversality theorem should be enough. Thanks! $\endgroup$ Commented Nov 24, 2014 at 23:54