📂Geometry

Definition of Intrinsic in Differntial Geometry

Definition¹

In differential geometry, a function that depends only on the coefficients of the first fundamental form $g_{ij}$, and not on the unit normal $\mathbf{n}$, is called intrinsic.

Explanation^{2 3}

If the coefficients of the Riemann metric $g_{ij}$ are known, then the length of curves on the surface and the area of the surface can be calculated without leaving the surface as follows can be calculated.

$$\text{length of } \alpha = \int_{a}^{b} \sqrt{ g_{ij} \alpha_{i}^{\prime} \alpha_{j}^{\prime} } dt = \int_{a}^{b} \sqrt{ E\left( \dfrac{d u_{1}}{dt} \right)^{2} + 2F\dfrac{d u_{1}}{dt}\dfrac{d u_{2}}{dt} + G\left( \dfrac{d u_{2}}{dt} \right)^{2}} dt$$

$$\text{area of } R = \iint _{Q} \sqrt{g} du_{1}du_{2} = \iint _{Q} \sqrt{EG-F^{2}} du_{1}du_{2}$$

This means that it can be determined through information on the tangent plane (coefficients of the first fundamental form) without using information outside the surface (for example, the unit normal $\mathbf{n}$). Therefore, things that can be calculated in this way are called intrinsic.

To view a surface $M$ from an intrinsic perspective means to think of $M$ as the entire space itself, and to view it from an extrinsic perspective means to consider it as a subspace of $M \subset \R^{3}$ and $\R^{3}$.

Examples of intrinsic things are as follows.

Examples

• Christoffel symbols
• Geodesic curvature

See Also

• Intrinsic spatial process in spatial statistics