[Differential Geometry] What does an Isometry on Smooth Manifolds preserve?

We will look into \(\varphi\), a diffeomorphic map, which is an invertible function that maps one differentiable manifold to another in a smooth way, where its inverse is also smooth. Now what are some things that can be preserved under these diffeomorphic maps? Let's look into what we call an isometry. If \(\varphi: S \rightarrow \overline{S}\) is an isometry between two smooth manifolds \(S\) and \(\overline{S}\), then for all points \(p \in S\) and all pairs of of tangent vectors \(v\), \(w\) in \(T_p(S)\),

\[ \left< v , w \right>_p = \left <d\varphi_p(v), d\varphi_p(w) \right>_{\varphi(p)}. \]

This means that the inner product of all tangent vectors at point \(p\) in surface \(S\) is preserved by the differential \(d\varphi\) of the map. If the inner product is preserved, then the first fundamental form, hence the coefficients of the first fundamental form are also preserved by the isometry from the following:

\[ 2I_p(v) = \left<v,v\right>_p = \left<d\varphi(v), d\varphi(v)\right>_{\varphi(p)} = 2I_{\varphi(v)}(d\varphi(v)). \] \[\therefore I_p(v) = I_{\varphi(v)}(d\varphi(v)). \]

Therefore, the first fundamental form, \( I_p(S)\) is also preserved under an isometry.

On the other hand, if we know that the first fundamental form is preserved under an isometry, then we can also conclude that the inner product of any pair of vectors on the tangent plane is preserved too, By polarisation, \[ 2\left<v,w\right>_p = I_p(v+w) - I_p(v) - I_p(w) = I_{d\varphi(p)}(v+w) - I_{d\varphi(p)}(v) - I_{d\varphi(p)}(w) = 2\left<d\varphi(v),d\varphi(w)\right>_{\varphi(p)}.\]

\[\therefore \left<v,w\right>_p = \left<d\varphi(v),d\varphi(w)\right>_{\varphi(p)}.\]

Note by Tasha Kim
11 months, 3 weeks ago

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