I got the idea that expansion of the universe is not to somewhere, it is just getting stretched of spacetime since a point of singularity. And I know that universe was calculated as flat (which means sum of angles of a triangle is exactly π ignoring local instrinsic curvatures caused by celestial masses) with a very small error rate. But still universe is possible to be hyperbolical or elliptical. If it is elliptical, when you travel on a straight route, you will eventually arrive at your starting point (we assume expansion of the universe is frozen). So, can we say that in that travel I am actually circumnavigating on the surface of a higher dimensional sphere and that means spacetime has an extrinsic curvature beside its intrinsic curvature.
If there was a higher-dimensional space in which the universe were embedded, then yes. But there is no need for any such space: the 3-sphere can exist happily on its own in 3 dimensions.
If this seems weird, take the simpler example of a cylinder. We’re used to thinking of the cylinder as a 3-dimensional volume, but the cylinder is sufficiently defined in 2 dimensions thus: take a 2-dimensional semi-infinite plane and glue together one pair of opposing edges. This “gluing” operation can be imagined as a topological rule defined on the plane, and needs no higher dimension in order to be realized.