There are Only Three Spatial Dimensions

The direction of a straight line or segment can only be reversed by inverting it. This requires at least one additional spatial dimension. An oriented triangle can only be flipped with the help of another dimension. If you place a normal to this triangle, you have another point for an oriented tetrahedron. But this cannot be flipped. L-amino acids do not automatically convert into d-amino acids.*
The fourth dimension of spacetime, time, is of no help here. A triangle can be moved arbitrarily within its plane over time. Without the help of the third spatial dimension, its orientation cannot be changed. As you can see, there are only three spatial dimensions. The relativity of space and time is completely described by the change in spatial scale, according to the general law of relativity.
It is not necessary to consider space as curved space. However, differentiable functions can be determined coordinate-free, even with internal curvature. In the Schwarzschild metric, the potential can be replaced with the "snow globe potential".** This metric is derived from scale changes and has no singularity (for R>0).
A curved space is defined as a manifold with dimension possibly smaller than a surrounding Euclidean or non-Euclidean space, whose tangent space and its normal space complement each other to form a (rotated) coordinate system of the surrounding space. For example, a sphere is not curved in this way, only its surface. If there is no normal space, it can be supplemented with a virtual dimension.

Ludwig Resch

*This refers only to physical, not mathematical dimensions.
**Only applies to the maximum length scale.