## Reimannian Geometry

### Extrinsic vs Intrinsic Geometry

Prior to Riemann, geometry was studied from an

*extrinsic*point of view, i.e, lines, curves and surfaces were considered to be embedded in a Euclidean space which adhered to the*usual*notions of distances, angles, translation and rotations. But Gauss showed through his*Theorema Egregium,*that Gaussian curvature is intrinsic, that is it can be computed by measuring angles, distances and their rates on the surface without considering the embedding itself (recall that $\kappa=\det(-I^{-1}II)$). That means that the embedding can often be ignored while studying certain geometric properties which are*intrinsic*.
But this means, that properties like Gaussian curvature are invariant to

*isometry*, i.e. deformations that bend the surface without stretching or contraction since local geometric properties are preserved. This in turn implies that a sphere could never be unwrapped onto a plane surface without stretching or contraction of the surface because a plane has a 0 Gaussian curvature whereas a sphere has a Gaussian curvature of 1. This basically says you cannot map all of earth in a single*chart*without messing up distances between far away cities or areas of continents? So what do we do? We build an*atlas*consisting of multiple*charts*where each*chart*provides a good approximation of local distances and angles in a specific location on the earth. The motivation behind Reimannian geometry and the concept of*Manifold*is similar.
Now let us define a few terms in a top down fashion - earlier definitions use terms or details that are defined later on.

#### Riemannian Geometry

Branch of differential geometry that deals with

*Riemannian Manifolds*.#### Riemannian Manifold

A smooth manifold $M$ equipped with an inner product $g_p$ (

*Riemannian metric*such as $I$) on tangent space (vector space of tangents at a point) $T_pM$ at each point $p$ that varies smoothly from point to point. Meaning that if $X$ and $Y$ are vector fields on $M$, then $g_p(X(p),Y(p))$ is a smooth function.#### Manifold

A topological space that locally resembles a Euclidean space. Meaning that each point of the $n$-dimensional manifold is

*homeomorphic*to a Euclidean space of the same dimension.#### Homeomorphism

A function $\phi$ between topological spaces $U$ and $\Omega$ in $\mathbb{R}^n$ such that

- $\phi$ is one-to-one and onto
- $\phi$ is continuous
- $f^{-1}$ is continuous

#### $C^p$ Atlas

A $C^p$ atlas on $M$ is given by

- An open cover $\{U_i | i \in I\}$ of $M$
- A family of homeomorphisms $\phi_i : U_i \rightarrow \Omega_i$
- For all $i,j\in I$, $\phi_j \circ \phi_i^{-1}$ is a $C^p$ diffeomorphism (differentiable homeomorphism) from $\phi_i(U_i \cap U_j)$ to $\phi_j(U_i \cap U_j)$

$(U_i, \phi_i)$ together define a $chart$.