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Assume that we have a Riemannian immersion of an $n$-dimensional Riemannian manifold into $\mathbb R^{n+1}$. If $n ≥ 3$, then does $M$ necessarily have non-negative sectional curvature?(I know if $n=2$, this conclusion is wrong. But I do not know whether this holds in higher dimension.)

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up vote 2 down vote accepted

You said, "I know if $n=2$, this conclusion is wrong." That is to say, you know a Riemannian immersion of a $2$-dimensional Riemannian manifold $M$ into $\mathbb{R}^3$ such that $M$ does not have non-negative sectional curvature.

Now take the product $M\times\mathbb{R}^k$ where $k\geq 1$. Then there exists a Riemannian immersion of the $(k+2)$-dimensional Riemannian manifold $M\times\mathbb{R}^k$ into $\mathbb{R}^{k+3}$ such that $M\times\mathbb{R}^k$ does not have non-negative sectional curvature.

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