### MATH 742 : Geometric Analysis

Fall 2019
Here is the syllabus for the course in pdf format.

#### Announcements

We are meeting in Math 0103 starting Monday, August 26.

Office hours Mondays 2-2:50, or by appointment, in 4412.

#### Problem Sets

• HW 1, due 9/11 FINAL

expand collapse

1. Let φ be a coordinate chart on an n-dimensional manifold M defined at a point p. Define tangent vectors by $$v_i(f) = \frac{\partial (f \circ \varphi^{-1})}{\partial x_i} \qquad f \in C^\infty (M)$$ Show that {v1, ..., vn} form a basis for Tp M, the tangent space to M at p.
2. In this problem you are asked to compute tangent spaces of manifolds embedded in Rn+1. In this case, elements of Tp M can equivalently be viewed as vectors in Rn+1, based at p. It is these subspaces of Rn+1 which you are asked to compute below.

a) Show that Tp Sn = p.

b) Recall the notation for the nondegenerate quadratic forms on Rn of index (p,q), where p+q = n: $$Q_{p,q}({\bf x}) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - x_n^2$$ Consider the manifolds $$S^{p,q} = \{ {\bf x} \in {\bf R}^{n+1} \ : Q_{p+1,q}({\bf x}) = 1 \}$$ and $$H^{p,q} = \{ {\bf x} \in {\bf R}^{n+1} \ : \ Q_{p,q+1}({\bf x}) = -1 \}$$ Show that Tp Sp,q = p and similarly for Hp,q. Thus both are pseudo-Riemannian manifolds of index (p,q).

3. For a smooth path γ:[a,b] → M,

a) assume that M is Riemannian, and let $$\alpha(t) = \int_a^t \left\| \dot{\gamma}(t) \right\| \ dt$$ Then show that γ composed with α-1 is parametrized by arc length;

b) assume that M is pseudo-Riemannian, and show that a smooth timelike path admits a parametrization by proper time.
4. Compute the Euler-Lagrange equations for the hyperbolic plane in the upper-half plane model $$g_{(x,y)} = \frac{dx^2 + dy^2}{y^2}$$ a) for the energy Lagrangian;

b) for the length Lagrangian.
5. In lecture we computed the matrix of leading coefficients in the Euler-Lagrange equations for the length Lagrangian on an n-dimensional Riemannian manifold: $$\left( a_{km} \right) = \left( \frac{g_{km}}{\left\| \dot{\gamma} \right\|} - \frac{\left( \Sigma_{\ell} g_{k\ell} \dot{\gamma}^\ell \right) \left( \Sigma_{\ell} g_{m\ell} \dot{\gamma}^\ell \right) }{\left\| \dot{\gamma} \right\|^3} \right)$$ Show that the rank of this matrix is n-1.
• HW 2, due 9/25

1. Let γ be a smooth curve in a semi-Riemannian manifold M. Given two coordinate charts $$\varphi = (x_1, \ldots, x_n) \qquad \bar{\varphi} = (\bar{x}^1, \ldots, \bar{x}^n)$$ use our transformation rules for Christoffel symbols to show that $$\varphi \circ \gamma = (\gamma^1, \ldots, \gamma^n)$$ satisfies the geodesic equations in the first coordinates if and only if $$\bar{\varphi} \circ \gamma = (\bar{\gamma}^1, \ldots, \bar{\gamma}^n)$$ satisfies the geodesic equations in the second coordinates.
2. (Postnikov Exer. 12.3) For the Lagrangian $$\mathcal{L} = \dot{x}^2 + \dot{y}^2$$ on the R2, show that the path x = at, y = 0 is an extremal. Let A be the point (a,0), a > 0, on this extremal, and let O be the origin. The integral $$\int_0^1 (\dot{x}^2 - \dot{y}^2) dt$$ over the segment OA of this extremal is equal to a2. Compute that the action over the broken line OBA (correspondingly parametrized), where B is the point (a/2,b/2), b>0, is equal to a2 - b2. Therefore, the segment OA is not a minimum curve, and there is no minimum curve connecting the points O and A for this Lagrangian.
3. Show that a timelike geodesic in a Lorentzian manifold locally maximizes proper time.