If a non-orientable manifold (like a Möbius strip) is embedded in 3D space and continuously deformed, which consequence relating to vector fields is most likely?

A)Vector fields remain divergence-free always

B)Parallel transport preserves vector direction

C)Global vector fields must have singularities✓

D)Intrinsic curvature stays constant everywhere

💡 Explanation

Because a non-orientable manifold lacks a consistent normal vector field, any attempt to define a smooth, non-vanishing vector field across the entire surface results in singularities due to the Hairy Ball Theorem. Therefore, singularities must occur, rather than parallel transport being preserved, because non-orientability prevents consistent directionality.

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If a non-orientable manifold (like a Möbius strip) is embedded in 3D space and continuously deformed, which consequence relating to vector fields is most likely?

💡 Explanation

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