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Antarres

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I'm not sure this is a differential geometry question, but I think it is.

In general when we have a hypersurface(or in case of 3D space just a surface) it is defined with an equation ##f(x^a)=0## for some function ##f##. Then the normal vector is the gradient of this function, if we want an unit vector we normalize it.

But what if this gradient is zero on the whole surface? How do we define the normal vector then? I called this surface 'extremal' because in black hole theory, horizons have this property in extremal solutions. But horizons are null hypersurfaces and we can avoid this for example using Killing vectors(because rigidity theorems tell us a Killing vector generates this surface as well).

But what do we do for an arbitrary surface?

Edit: By mistake I posted this in differential equations instead of differential geometry... I apologize for the mistake, the question can be moved to differential geometry maybe?

In general when we have a hypersurface(or in case of 3D space just a surface) it is defined with an equation ##f(x^a)=0## for some function ##f##. Then the normal vector is the gradient of this function, if we want an unit vector we normalize it.

But what if this gradient is zero on the whole surface? How do we define the normal vector then? I called this surface 'extremal' because in black hole theory, horizons have this property in extremal solutions. But horizons are null hypersurfaces and we can avoid this for example using Killing vectors(because rigidity theorems tell us a Killing vector generates this surface as well).

But what do we do for an arbitrary surface?

Edit: By mistake I posted this in differential equations instead of differential geometry... I apologize for the mistake, the question can be moved to differential geometry maybe?

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