Differential Geometry

Prerequisites

A vector space $V$ (over $R$) is a set with two operations:

• Vector addition
• Scalar multiplication

such that $(V,+)$ is an abelian group and, for all $a,b\in R$, $v,w\in V$

• $a(bv)=(ab)v$
• $(a+b)v=av+av$
• $a(v+w)=av+aw$
• $1v=v$

Let $V$ be a vector space. A subspace $W$ of $V$ is a subset of $W\subset V$ such that, $av+bw\in W$whenever $v,w\in W$ and $a,b\in R$.

Let $V$ and $W$ be vector spaces. A function $T:V\to W$ is a linear map (or linear transformation) if: $T(av+bw)=aT(v)+bT(w)$for all $v,w\in V$ and $a,b\in R$.

Proposition: If $T:V\to W$ is a linear map, then

• $\text{Ker}(T)=\{v\in V:T(v)=0\}$ is a subspace of $V$ and
• $\text{Im}(T)$ = $\{w\in W:w=T(v)\text{ for some }v\in V\}$ is a subspace of $W$.

Example:

Let $V\in C^{\infty }(R)$. If $W$ is a set of all polynomials, then $W$ is a subspace. For $k\ge 1$, if $W_k$ is the set of all polynomials of degree less than or equal to $k$, then $W_k$ is a subspace. For $k\ge 1$, the $k$-th derivative gives a linear map $D^k:V\to V$: $D^k(f)(t)=f^{(k)}(t)$for any $f\in V$ and $t\in R$. We have $\text{Ker}(D^k)=W_{k-1}$

Let $\{v_1,\dots ,v_k\}\subset V$ be a finite subset. A linear combination of $v_1,\dots ,v_k$ is an element of $V$ of the form $c_1{v}_1+\dots +c_k{v}_k$, for some $c_1,\dots ,c_k\in R$. The span of a subset $S\subset V$ is the set: $\text{span}(S)=\{w:w\text{ is a linear combination of elements from }S\}$ Easy facts:

$\text{span}(S)$ is a subspace of $V$. $S$ is a subspace if and only if $\text{span}(S)=S$.

$v_1,\dots ,v_k\in V$ are said to be linear independent if the following conditions: $c_1{v}_1+\dots +c_k{v}_k=0$for some $c_1,\dots ,c_k\in R$ then necessarily $c_1=\dots =c_k=0$.

If $v_1,\dots ,v_k$ are not linearly independent, we say that they are linearly dependent.

The set $\{v_1,\dots ,v_n\}$ is a basis for $V$ if

• $\text{span}(v_1,\dots ,v_n)=V$ and
• $v_1,\dots ,v_n$ are linearly independent.

If $\{v_1,\dots ,v_n\}$ and $\{w_1,\dots ,w_m\}$ are bases for $V$, then $n=m$.

Corollary: Let $V$ be a n-dimensional vector space and let $S=\{v_1,\dots ,v_n\}\subset V$. If either of the following holds, then $S$ is a basis.

• $S$ is linearly independent or,
• $\text{span}(S)=V$

Example: $C^{\infty }(R)$ does not have a basis.

The rank-nullity theorem:

Let $V$ and $W$ be finite-dimensional vector spaces and $T:V\to W$ a linear map. Then, $\text{dim}(\text{Ker}(T))+\text{dim}(\text{Im}(T))=\text{dim}(V)$

Let a function $f:(a,b)\to R$, suppose $p\in (a,b)$ we say that, $f$ is differentiable at $p$, with derivative $f^{\prime }(p)$ if ${\lim }_{h\to 0}\frac{f(p+h)-f(p)}{h}=f^{\prime }(p)$ Now consider $f:U\to R$, where $U\subseteq R^n$, $n\ge 1$ is an open subset. Let $p\in U$, then difference quotient will not work.

Rewrite, ${\lim }_{h\to 0}\left|\frac{f(p+h)-f(p)-f^{\prime }(p)h}{h}\right|=0$ This works for $n\ge 2$. Observe that $h\to f^{\prime }(p)h$ is a linear map from $R$ to $R$. This motivates that $f:U\to R$ is differentiable at $p$ if there exists a linear map $L:R^n\to R$ such that ${\lim }_{h\to 0}\frac{|f(p+h)-f(p)-L(h)|}{||h||}=0$

Proposition: If such a linear map exists, then it is unique.

Definition: $L$ is said to be the derivative (Frechet derivative) of $f$ at $p$.

Now, consider $f:U\to R^m$, $U\subseteq R^n$. Let $p\in U$, we say that $f$ is differentiable at $p$ if one of the following equivalent conditions holds:

• There is a linear map $L:R^n\to R^m$ such that ${\lim }_{h\to 0}\frac{||f(p+h)-f(p)-L(h)||}{||h||}=0$
• If we write $f=(f_1,\dots f_m)$. Then each $f_i$ is differentiable at $p$. Note that $f_i:U\to R$

We denote $L$ by $d{f}_p$ or $D{f}_p$.

Proposition: Let $f:U\to R^m$ and $p\in U$. If $\frac{\partial f_i}{\partial x_j}$, $1\le i\le m$, $1\le j\le n$ exist in neighborhood of $p$ and they are continuous there, then $f$ is differentiable at $p$.

In fact, the matrix of $d{f}_p$ (with respect to the standard bases of $R^n$ and $R^m$) is given by: $[d{f}_p{]}_{ij}=\frac{\partial f_i}{\partial x_j}$ Also called the Jacobian matrix of $f$ at $p$.

Recall the partial derivatives of a function $f:U\to R$. Let $p\in U$, $\frac{\partial f}{\partial x_i}(p):={\lim }_{t\to 0}\frac{f(p+t{e}_i)-f(p)}{t}$ Here $e_1=(1,\dots ,0),\dots ,e_n=(0,\dots ,1)$ is the standard basis of $R^n$.

Now, let $v\in R^n$. If the limit $v(f):={\lim }_{t\to 0}\frac{f(p+tv)-f(p)}{t}$exists, then we call it the directional derivative of $f$ along $v$ at $p$.

Partial derivatives vs directional derivatives: $e_i(f)=\frac{\partial f}{\partial x_i}(p)$

Let $f:U\to R$ and $p\in U$, $v\in R^n$. Suppose that $f$ is differentiable at $p$. Then we have, $d{f}_p(v)=v(f)$

Chain rule:

Let $R^n\stackrel{f}{\to }{R}^m\stackrel{g}{\to }{R}^k$. If $f$ is differentiable at $p\in R^n$, and $g$ is differentiable at $q=f(p)\in R^m$, then $h=g\circ f$ is differentiable a $p$ and $d{h}_p=d{g}_{f(p)}\circ d{f}_p$

Manifolds

Idea: spaces which locally resemble open sets in Euclidean space.

The first requirement for transferring the ideas of calculus to manifolds is some notion of smoothness.

Definition:

Topological space is a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighborhoods for each point that satisfy some axioms formalizing the concept of closeness. A topological space is the most general type of mathematical space that allows for the definition of limits, continuity, and connectedness.

A homeomorphism $f:X\to Y$ is a continuous, bijective map such that $f^{-1}:Y\to X$ is continuous.

Suppose $M$ is a topological space. We say that $M$ is a topological manifold of dimension $n$ or a topological n-manifold if it has the following properties:

$M$ is a Hausdorff space: for every pair of distinct points $p,q\in M$, there are disjoint open subsets $U,V\subseteq M$ such that $p\in U$ and $q\in V$.

$M$ is second-countable: there exists a countable basis for the topology of $M$.

$M$ is locally Euclidean of dimension $n$: each point of $M$ has neighborhood that is homeomorphic to an open subset of $R^n$.

A topological manifold of dimension $n$ is Hansdoff, second countable topological space $X$ such that every point $p\in X$ has a neighborhood $U$ which is homeomorphic to an open subset of $R^n$. Any Hansdoff, second countable, locally Euclidean topological space $X$ is metricable. In other words, there is a distance function $d$ on $X$ such that the topology induced by $d$ coincides with the topology on $X$.

The pair of homeomorphic $\varphi :U\subseteq X\to V\subseteq R^n$ i.e. $(U,\varphi )$ is called a chart. The collection of charts $\{(U_{\alpha },{\varphi }_{\alpha }){\}}_{\alpha }$ which covers $X$ is called an atlas.

Alternative definition of topological manifolds of dimension $n$: $X$ is a separable metric space which is locally Euclidean of dimension $n$.

Coordinate charts:

Let $M$ be a topological n-manifold. A coordinate chart (or just a chart) on $M$ is a pair $(U,\varphi )$, where $U$ is an open subset of $M$ and $\varphi :U\to \hat{U}$ is a homeomorphism from $U$ to an open subset $\hat{U}\to \varphi (U)\subseteq R^n$. By definition of a topological manifold, each point $p\in M$ is contained in domain of some chart $(U,\varphi )$. If $\varphi (p)=0$, we say that the chart is centered at p. If $(U,\varphi )$ is any chart whose domain contains $p$, it is easy to obtain a new chart centered at $p$ by subtracting the constant vector $\varphi (p)$. Given a chart $(U,\varphi )$, we call the set $U$ a coordinate domain, or a coordinate neighborhood of each of its points.

The map $\varphi$ is called a local coordinate map and the component functions $(x^1,\dots ,x^n)$ of $\varphi$, defined by $\varphi (p)=(x^1(p),\dots ,x^n(p))$, are called local coordinates on $U$. We sometimes denote the chart by $(U,(x^i))$.

Examples:

Graph of a continuous functions Spheres

Smooth structures:

To make sense of derivatives of real-valued maps between manifolds, we need to introduce a new kind of manifold called a smooth manifold. If $U$ and $V$ are open subsets of Euclidean spaces $R^n$ and $R^m$, respectively, a function $F:U\to V$ is said to be smooth $C^{\infty }$ if each of its component functions has continuous partial derivatives of all orders. If in addition $F$ is bijective and has a smooth inverse map, it is called a diffeomorphism.

Consider an arbitrary topological n-manifold $M$. Each point in $M$ is in the domain of a coordinate map $\varphi :U\to \hat{U}\subseteq R^n$. A plausible definition of smooth function on $M$ would be to say that $f:M\to R$ is smooth if and only if the composite function $f\circ {\varphi }^{-1}:\hat{U}\to R$ is smooth in the sense of ordinary calculus. But this will make sense only if this property is independent of the choice of coordinate chart. To guarantee this independence, we will restrict our attention to smooth charts.

Let $M$ be a topological n-manifold. If $(U,\varphi ),(V,\psi )$ are two charts such that $U\cap V\ne \varphi$, the composite map $\varphi \circ {\psi }^{-1}:\varphi (U\cap V)\to \psi (U\cap V)$ is called the transition map from $\varphi$ to $\psi$. Two charts $(U,\varphi )$ and $(V,\psi )$ are said to be smoothly compatible if either $U\cap V=\varphi$ or the transition map $\varphi \circ {\psi }^{-1}$ is a diffeomorphism. We define an atlas for $M$ to be a collection of charts whose domains cover $M$. An atlas $A$ is called a smooth atlas if any two charts in $A$ are smoothly compatible with each other.

Our plan is to define a smooth structure on $M$ by giving a smooth atlas, and to define a function $f:M\to R$ to be smooth if and only if $f\circ {\varphi }^{-1}$ is smooth in the sense of ordinary calculus for each coordinate chart $(U,\varphi )$ in the atlas. But, in general, there will be many possible atlases that give same smooth structure, in that they all determine the same collection of smooth functions on $M$. We make the following definition: a smooth atlas $A$ on $M$ is maximal if it is not properly contained in any larger smooth atlas. This means that any chart that is smoothly compatible with every chart in $A$ is already in $A$. If $M$ is a smooth manifold, any chart $(U,\varphi )$ contained in the given maximal smooth atlas is called a smooth chart.

Note: Two smooth atlases for $M$ determine the same smooth structure if and only if their union is a smooth atlas.

Examples:

Open subset of Euclidean spaces Finite dimensional vector spaces

Smooth maps

Let $M,N$ be smooth manifolds, and let $F:M\to N$ be any map. We say that $F$ is a smooth map if for every $p\in M$, there exists smooth charts $(U,\varphi )$ containing $p$ and $(V,\psi )$ containing $F(p)$ such that $F(U)\subseteq V$ and the composite map $\psi \circ F\circ {\varphi }^{-1}$ is smooth from $\varphi (U)$ to $\psi (V)$.

Tangent vectors and tangent spaces:

In order to make sense of calculus on manifolds, we need to introduce the tangent space to a manifold at a point. Any geometric tangent vector $v_a\in R_a^n$ yields a map $D_v{|}_a:C^{\infty }(R^n)\to R$, which takes the directional derivative in the direction $v$ at $a$: $D_v{|}_a(f)=D_{v}f(a)=\frac{d}{dt}f(a+tv){|}_{t=0}$ This operator is linear over $R$ and satisfies the product rule: $D_v{|}_a(fg)=f(a)D_v{|}_{a}g+g(a)D_v|af$

Let $M$ be a n-manifold and let $p\in M$. A linear map $v:C^{\infty }(M)\to R$ is called a derivation at $p$ if it satisfies: $v(fg)=f(p)vg+g(p)vf\text{ for all }f,g\in C^{\infty }(M)$The set of derivations of $C^{\infty }(M)$ at $p$, denoted by $T_{p}M$, is a vector space called the tangent space to $M$ at $p$. An element of $T_{p}M$ is called a tangent vector at $p$.

Often it is useful to consider the set of all tangent vector at all points of a manifold. Give a smooth manifold $M$, we define the tangent bundle of $M$, denoted by $TM$, to be disjoint union of the tangent spaces at all points of $M$.

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A vector field $X$ on $M$ is an assignment of a tangent vector $X_p\in T_{p}M$ to every $p\in M$. Let $M$ be a smooth manifold, and let $X:M\to TM$ be a rough vector field. If $(U,(x^i))$ is any smooth coordinate chart on $M$, then the restriction of $X$ to $U$ is smooth if and only if its component functions with respect to this chart are smooth.

Example: If $(U,(x^i))$ is any smooth chart on $M$, the assignment: $p\to {\frac{\partial }{\partial x^i}|}_p$ determines a vector field on $U$, called the ith coordinate vector field.

The primary objects associated with smooth vector fields are their integral curves, which are smooth curves whose velocity at each point is equal to the value of the vector field there. The collection of all integral curves of a given vector field on a manifold determines a family of all integral curves of given vector field on a manifold determines a family of diffeomorphisms of (open subsets of) the manifold, called flow.

Suppose $M$ is a smooth manifold. If $\gamma :J\to M$ is a smooth curve, then for each $t\in J$, the velocity vector ${\gamma }^{\prime }(t)$ is a vector in $T_{\gamma (t)}M$. We find a way backwards: given a tangent vector at each point, we seek a curve whose velocity at each point is equal to the given vector there. If $V$ is a vector field on $M$, an integral curve of $V$ is differentiable curve $\gamma :J\to M$ whose velocity at each point is equal to the value of $V$ at that point: ${\gamma }^{\prime }(t)=V_{\gamma (t)}\text{ for all }t\in J$ If $0\in J$, the point $\gamma (0)$ is called the starting point of $\gamma$.

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Flows: todo

Tensors:

for covectors $\omega \otimes \eta (v_1,v_2):V\times V\to R=\omega (v_1)\eta (v_2)$ is a multilinear map.

Riemannian Metrics

To define geometric objects such as lengths and angles on a vector space, one uses an inner product. For manifolds, the appropriate structure if Riemannian metric, which is essentially a choice of inner product on each tangent space, varying smoothly from point to point. A choice of Riemannian metric allows us to define geometric concepts such as lengths, angles, and distances on smooth manifolds. A Riemannian metric on $M$ is a smooth symmetric covariant 2-tensor field on $M$ that is positive definite at each point. A Riemannian manifold is a pair $(M,g)$, where $M$ is a smooth manifold and $g$ is a Riemannian metric on $M$. If $g$ is a Riemannian metric on $M$, then for each $p\in M$, the 2-tensor $g_p$ is an inner product on $T_{p}M$. b