I did not understand the first question

Question 1 Are there manifolds with the property that each connection on is never flat?

Because one of course can construct, on any manifold, a connection which is somewhere flat and somewhere not flat.

But of course there are manifolds that do not admit flat affine connections, the simples example is the sphere of any dimension $n>1$. Actually, in dimension 2 the answer (obtained in 1950th , I believe by Kuiper) is as follows: any nonclosed surface admits a flat torsionfree affine connection, and closed surfaces of zero euler characteristic and only them, admit flat torsionfree connection.

As mentioned by Julian Rosen, the answer on the 3rd question is positive and is in the mathoverflow discussion he mentioned

Answer to Question 2 is positive, a simplest example is $S^2 \times S^1$ with the following affine connection: take $R^3\setminus \{0\}$ and quotient it with respect to the group $Z$ acting by $x\mapsto 2 x$. Since the standard flat connection is invariant with respect to the action, the quotient which is $S^2 \times S^1$ carries a flat connection. It has no flat metric of course.

P.S. You question was studied, and the answers I was given were obtained in the framework of the so-called affine structures. Affine structure on a manifold such that the transition maps are affine mappings. The existence of an affine structure is equivalent to the existence of a flat torsionfree affine connection.

I did not understand the first question

Question 1 Are there manifolds with the property that each connection on is never flat?

Because one of course can construct, on any manifold, a connection which is somewhere flat and somewhere not flat.

But of course there are manifolds that do not admit flat affine connections, the simples example is the sphere of any dimension $n>1$. Actually, in dimension 2 the answer (obtained in 1950th , I believe by Kuiper) is as follows: any nonclosed surface admits a flat torsionfree affine connection, and closed surfaces of zero euler characteristic and only them, admit flat torsionfree connection.

As mentioned by Julian Rosen, the answer on the 3rd question is positive and is in the mathoverflow discussion he mentioned

Answer to Question 2 is positive, a simplest example is $S^2 \times S^1$ with the following affine connection: take $R^3\setminus \{0\}$ and quotient it with respect to the group $Z$ acting by $x\mapsto 2 x$. Since the standard flat connection is invariant with respect to the action, the quotient which is $S^2 \times S^1$ carries a flat connection. It has no flat metric of course.

P.S. You question was studied, and the answers I was given were obtained in the framework of the so-called affine structures. Affine structure on a manifold is an atlas such that the transition maps are affine transformations, i.e., are given by the formula $x\mapsto Ax+ B$ for a nondegenerate matrix $A$ and a vector $b$. The existence of an affine structure is equivalent to the existence of a flat torsionfree affine connection.

I did not understand the first question

Question 1 Are there manifolds with the property that each connection on is never flat?

Because one of course can construct, on any manifold, a connection which is somewhere flat and somewhere not flat.

But of course there are manifolds that do not admit flat affine connections, the simples example is the round sphere. Actually, in dimension 2 the answer (obtained in 1950th , I believe by Kuiper) is as follows: any nonclosed surface admits a flat affine connection, and closed surfaces of zero euler characteristic and only them, admit flat connection.

As mentioned by Julian Rosen, the answer on the 3rd question is positive and is in the mathoverflow discussion he mentioned

Answer to Question 2 is positive, a simplest example is $S^2 \times S^1$ with the following affine connection: take $R^3\setminus \{0\}$ and quotient it with respect to the group $Z$ acting by $x\mapsto 2 x$. Since the standard flat connection is invariant with respect to the action, the woutient which is $S^2 \times S^1$ carries a flat connection. It has no flat metric of course.

P.S. You question was studied, and the answers I was given were obtained in the framework of the so-called affine structures. Affine structure on a manifold such that the transition maps are affine mappings. The existence of affine structure is equivalent to the existence of an affine connection.

I did not understand the first question

Question 1 Are there manifolds with the property that each connection on is never flat?

Because one of course can construct, on any manifold, a connection which is somewhere flat and somewhere not flat.

But of course there are manifolds that do not admit flat affine connections, the simples example is the sphere of any dimension $n>1$. Actually, in dimension 2 the answer (obtained in 1950th , I believe by Kuiper) is as follows: any nonclosed surface admits a flat torsionfree affine connection, and closed surfaces of zero euler characteristic and only them, admit flat torsionfree connection.

As mentioned by Julian Rosen, the answer on the 3rd question is positive and is in the mathoverflow discussion he mentioned

Answer to Question 2 is positive, a simplest example is $S^2 \times S^1$ with the following affine connection: take $R^3\setminus \{0\}$ and quotient it with respect to the group $Z$ acting by $x\mapsto 2 x$. Since the standard flat connection is invariant with respect to the action, the quotient which is $S^2 \times S^1$ carries a flat connection. It has no flat metric of course.

P.S. You question was studied, and the answers I was given were obtained in the framework of the so-called affine structures. Affine structure on a manifold such that the transition maps are affine mappings. The existence of an affine structure is equivalent to the existence of a flat torsionfree affine connection.