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 $chart$.