Manifolds

Manifolds Type

A \(n\)-dimensional manifold \(M\) is made up of points with at least one coordinate system. A coordinate system is a one-to-one mapping that maps \(\mathbb{R}^n \mapsto M\) and all points \(p\in M\) lie in the domain of its coordinate system. The coordinate transformation between its coordinate systems must be a smooth as well.

• A manifold is a topological space that is locally equivalent to the Euclidean space.
• A differentiable manifold is a continuous and differentiable manifold
• A Riemannian manifold is a manifold with a symmetric (0,2) tensor that acts as a metric tensor on each point of the manifold and the signature of the metric tensor are all positive
• A psuedo-Riemannian manifold is a manifold with a symmetric (0,2) tensor that acts as a metric tensor on each point of the manifold and the signature of the metric tensor is (-,+,...,+)

Local-Flatness Theorem

By locally flat, we mean that in a neighborhood of a point \(P\), if the radius \(\epsilon\) of the neighborhood is small enough, one can always find a set of coordinate system such that \begin{align} g_{\alpha\beta}(P) &= \eta_{\alpha\beta}, \qquad \forall \alpha,\beta\\ \frac{\partial}{\partial x^\gamma}g_{\alpha\beta}(P)&=0, \qquad \forall \alpha,\beta,\gamma, \end{align} where \(\eta_{\alpha\beta}\) is the component of the metric tensor of a flat Euclidean space. Given a set of \(\{x^\alpha\}\) coordinate system, one wants to find a set of \(\{x^\alpha{'}\}\) coordinate system such that the above conditions are satisfied. At a point \(P\), the two sets of coordinates are related by the transformation matrix $$\Lambda^\alpha_{\mu'} = \frac{\partial x^\alpha}{\partial x^{\mu'}}.$$ Perform Taylor expansion about the point \(P\), \begin{align} \Lambda^\alpha_{\mu'}(\vec{x}) &= \Lambda^\alpha_{\mu'}(P) + (x^{\gamma'}-x^{\gamma'}_0)\frac{\partial \Lambda^\alpha_{\mu'}(P)}{\partial x^{\gamma'}} \\ &\quad+ \frac{1}{2}(x^{\gamma'}-x^{\gamma'}_0)(x^{\lambda'}-x^{\lambda'}_0)\frac{\partial^2\lambda^\alpha_{\mu'}(P)}{\partial x^{\lambda'}\partial x^{\gamma'}}+...\\ &=\Lambda^\alpha_{\mu'}|_P + (x^{\gamma'}-x^{\lambda'}_0)\frac{\partial^2x^\alpha}{\partial x^{\gamma'}\partial x^{\gamma'}}|_P \\ &\quad+ \frac{1}{2}(x^{\gamma'}-x^{\gamma'}_0)(x^{\lambda'}-x_0^{\lambda'})\frac{\partial^3x^\alpha}{\partial x^{\lambda'}\partial x^{\gamma'}\partial x^{\gamma'}}|_P +... \end{align} Similarly for the metric, \begin{align} g_{\alpha\beta}(\vec{x}) &= g_{\alpha\beta}|_P + (x^{\gamma'}-x^{\gamma'}_0)\frac{\partial g_{\alpha\beta}}{\partial x^{\gamma'}}|_P \\ &\quad + \frac{1}{2}(x^{\gamma'}-x^{\gamma'}_0)(x^{\lambda'}-x^{\lambda'}_0)\frac{\partial^2g_{\alpha\beta}}{\partial x^{\lambda'}\partial x^{\gamma'}}|_P. \end{align} Substitute into the metric transformation equation $$g_{\mu'\nu'} = \Lambda^\alpha_{\mu'}\Lambda^\beta_{\nu'}g_{\alpha\beta},$$ we have \begin{align} g_{\mu'\nu'}(\vec{x}) &= \Lambda^{\alpha}_{\mu'}|_P\Lambda^\beta_{\nu'}|_Pg_{\alpha\beta}|_P \\ &\quad +(x^{\gamma'}-x^{\gamma'}_0)(\lambda^\alpha_{mu'}|_P\Lambda^\beta_{\nu'}|_Pg_{\alpha\beta,\gamma}|_P\\ &\quad + \Lambda^\alpha_{\mu'}|_P ga_{\alpha\beta}|_P \frac{\partial ^2x^\beta}{\partial x^{\gamma'}x^{\nu'}}|_P\\ &\quad + \Lambda^\beta_{\nu'}|_P g_{\alpha\beta}|_P \frac{\partial^2 x^\alpha}{\partial x^{\gamma'}\partial x^{\mu'}}|_P)\\ &\quad +\frac{1}{2}(x^{\gamma'}-x^{\gamma'}_0)(x^{\lambda'}-x^{\lambda'}_0)(...)\\ \end{align} Since by definition, the metric tensor \(\eta_{\mu'\nu'}\) is symmetric, there are (nXn-n)/2+n variables. While the transformation matrix \(\Lambda^\alpha_{\mu'}\) has nXn free variables, the transformation matrix can be chosen such that $$\eta_{\mu'\nu'} = \Lambda^\alpha_{\mu'}|_P\Lambda^{\beta}_{\nu'}|_Pg_{\alpha\beta}|_P.$$ Differentiate \(g_{\mu'\nu'}(\vec{x})\), since $$\frac{\partial \Lambda^\alpha_{\mu'}}{\partial x^{\gamma'}}|_P = \frac{\partial^2x^\alpha}{\partial x^{\gamma'}\partial x^{\mu'}}|_P$$ is also symmetric as partial derivative commutes, we have (nXn-n)/2+n free variables just enough to fit the (nXn-n)/2+n variables such that $$g_{\alpha'\beta',\mu'}|_P=0.$$ That shows that it is possible to be locally flat if the manifold is infinitely differentiable. For psuedo-Riemannian space, \(\eta_{\mu'\nu'}\) will be the Minkowski metric instead.

Length

The length element of a curve is given by the metric $$dl = |g_{\alpha\beta}dx^\alpha dx^\beta|^{1/2}.$$ Then the length is given by $$l = \int |g_{\alpha\beta}dx^\alpha dx^\beta|^{1/2}.$$ Let the curve be parametrized by \(\lambda\), then the length is $$l=\int^{\lambda_1}_{\lambda_0}|g_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx\beta}{d\lambda}|d\lambda.$$ If \(\vec{V}\) is the tangent vector of the curve, then the components of \(\vec{V}\) are $$V^\alpha=\frac{dx^\alpha}{d\lambda}.$$ Then, the curve length is $$l = \int^{\lambda_1}_{\lambda_0}|\vec{V}\cdot\vec{V}|^{1/2}d\lambda$$

Volume

Let \(R\) be a bounded region with a coordinate system \(x^i\). Then, the volume of \(R\) is defined as $$\int ...\int \sqrt{|\det g_{ij}|}dx^1...dx^n,$$ over the region \(\{(x^i(p)|p\in R\}\), such that it is an invariant under the transformation of coordinate system.

Traditionally, we call the integral as area when \(R\) is 2D, and arclength for 1D.