In my situation, I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ must exhibit a full sphere of normal spaces. Suppose instead that $M / \sim$ has a handle. Can I always choose a loop around the handle that also implies that the normals of $M / \sim$ must vary by at least 90 degrees?
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1This is not a research question. I've voted to migrate it. – Ryan Budney Mar 24 '16 at 03:40
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Old question but this may help - https://math.stackexchange.com/questions/423282/normal-vector-to-a-sphere – Jacksonkr Dec 01 '21 at 19:44