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LSpice
LSpice
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As Anton writeswrites, this is unknown.

The main issue is that as of now, the only method we have of creating positively curved closed manifolds with non-trivial fundamental group is to start with a simply connected example and quotient by a free isometric action.

The only positively curved metrics on spheres which we understand well enough to carry this out are induced by invariant metrics on Lie groups. In particular, the only quotients which inherit positive curvature from this process are space forms.

If you weaken your question by changing "positive" to "non-negative", then we know a bit more. For example, Theorem G of

Grove, Karsten, and Wolfgang Ziller. “Curvature and Symmetry of Milnor Spheres.” Annals of Mathematics 152, no. 1 (2000): 331–67. arXiv version

asserts that all four oriented diffeomorphism types of $\mathbb{R}P^5$s admit metrics of non-negative sectional curvature. Since there are no exotic $5$-spheres, these examples meet your criteria (except for merely having non-negative sectional curvature.).

As Anton writes, this is unknown.

The main issue is that as of now, the only method we have of creating positively curved closed manifolds with non-trivial fundamental group is to start with a simply connected example and quotient by a free isometric action.

The only positively curved metrics on spheres which we understand well enough to carry this out are induced by invariant metrics on Lie groups. In particular, the only quotients which inherit positive curvature from this process are space forms.

If you weaken your question by changing "positive" to "non-negative", then we know a bit more. For example, Theorem G of

Grove, Karsten, and Wolfgang Ziller. “Curvature and Symmetry of Milnor Spheres.” Annals of Mathematics 152, no. 1 (2000): 331–67. arXiv version

asserts that all four oriented diffeomorphism types of $\mathbb{R}P^5$s admit metrics of non-negative sectional curvature. Since there are no exotic $5$-spheres, these examples meet your criteria (except for merely having non-negative sectional curvature.)

As Anton writes, this is unknown.

The main issue is that as of now, the only method we have of creating positively curved closed manifolds with non-trivial fundamental group is to start with a simply connected example and quotient by a free isometric action.

The only positively curved metrics on spheres which we understand well enough to carry this out are induced by invariant metrics on Lie groups. In particular, the only quotients which inherit positive curvature from this process are space forms.

If you weaken your question by changing "positive" to "non-negative", then we know a bit more. For example, Theorem G of

Grove, Karsten, and Wolfgang Ziller. “Curvature and Symmetry of Milnor Spheres.” Annals of Mathematics 152, no. 1 (2000): 331–67. arXiv version

asserts that all four oriented diffeomorphism types of $\mathbb{R}P^5$s admit metrics of non-negative sectional curvature. Since there are no exotic $5$-spheres, these examples meet your criteria (except for merely having non-negative sectional curvature).

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Jason DeVito - on hiatus
Jason DeVito - on hiatus
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As Anton writes, this is unknown.

The main issue is that as of now, the only method we have of creating positively curved closed manifolds with non-trivial fundamental group is to start with a simply connected example and quotient by a free isometric action.

The only positively curved metrics on spheres which we understand well enough to carry this out are induced by invariant metrics on Lie groups. In particular, the only quotients which inherit positive curvature from this process are space forms.

If you weaken your question by changing "positive" to "non-negative", then we know a bit more. For example, Theorem G of

Grove, Karsten, and Wolfgang Ziller. “Curvature and Symmetry of Milnor Spheres.” Annals of Mathematics 152, no. 1 (2000): 331–67. arXiv version

asserts that all four oriented diffeomorphism types of $\mathbb{R}P^5$s admit metrics of non-negative sectional curvature. Since there are no exotic $5$-spheres, these examples meet your criteria (except for merely having non-negative sectional curvature.)