## Unit normal to a radial graph over sphere

Consider is a domain in the sphere. is a radial graph over .

What is the unit normal to this radial graph?

Suppose is a smooth local frame on . Let be the covariant derivative on . Tangent space of consists of which are

In order to get the unit normal, we need some simplification. Let us assume are orthonormal basis of the tangent space of and . Then

Then we obtain an orthonormal basis of the tangent of

We are able to get the normal by projecting to this subspace

After normalization, the (outer)unit normal can be written

**Remark: **Guan, Bo and Spruck, Joel. *Boundary-value Problems on \mathbb{S}^n for Surfaces of Constant Gauss Curvature.*

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