Project Curriculum: A Guide to Some Topics to Help Follow Along¶

How to Use This Document¶

Each section corresponds to one paper/set of notes. For every topic listed, your task is threefold: 1. Understand it yourself to the point you can derive the key results from scratch 2. Write an explanation that a calculus-fluent freshman can follow, using concrete examples before abstract definitions 3. Verify your understanding at the checkpoints so that you can read the notebooks with clear understanding.

Part 1: Curves and Surfaces in \(\mathbb{R}^3\)¶

Goal: Establish the proper formulations of geometric objects we are working with.

Section 0: Prerequisites -- Linear Algebra¶

Inner Products and Projections - [] Define the dot product on \(\mathbb{R}^n\) as a bilinear, symmetric, positive definite form. State all three properties precisely. - [] Define the norm induced by the dot product: \(|v| = \sqrt{v \cdot v}\) - [] Define the angle between two vectors via \(\cos \theta = (u \cdot v)/(|u||v|)\). Derive this from the law of cosines. - [] State and prove: two vectors are orthogonal iff their dot product is zero - [] Define orthogonal projection of \(u\) onto \(v\): \(\operatorname{proj}_v(u) = (u \cdot v / v \cdot v)\, v\) - [] Define the component of \(u\) orthogonal to \(v\): \(u_\perp = u - \operatorname{proj}_v(u)\) - [] Key result: Write the projection onto a subspace \(W\) with orthonormal basis \(\{e_1, \ldots, e_k\}\): \(P_W(u) = \sum (u \cdot e_i) e_i\). Derive \(P_{W^\perp}(u) = u - P_W(u)\). This is the formula \(P_x v = v - (v \cdot x)x\) used in BM projection when \(x\) is the unit normal. - [] Define orthogonal complement \(W^\perp\) and prove \(\mathbb{R}^n = W \oplus W^\perp\)

Matrices as Linear Maps - [] State: every linear map \(T: \mathbb{R}^n \to \mathbb{R}^m\) has a matrix representation \([T]\) depending on the choice of basis - [ ] Define symmetric matrix (\(A = A^\top\)) and state the spectral theorem: every symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors - [ ] Define positive definite matrix: \(A\) is positive definite iff \(v^\top A v > 0\) for all \(v \neq 0\). Equivalently, all eigenvalues are positive. - [ ] Define a change-of-basis matrix \(P\) and how components of a vector transform: if \(v = \sum v^i e_i\) in basis \(\{e_i\}\) and \(\{\tilde{e}_j\}\) is a new basis, how do the components \(v^i\) change? (Contravariant transformation rule) - [ ] State how matrix representations of linear maps change under change of basis: \([T]_{\text{new}} = P^{-1} [T]_{\text{old}} P\). This is why eigenvalues are basis-independent.

Quadratic Forms - [ ] Define a quadratic form \(Q(v) = v^\top A v\) for symmetric \(A\) - [ ] Note: the first fundamental form (metric tensor) will be a quadratic form -- it assigns a positive number to every tangent vector, measuring its squared length

Section 1: Prerequisites -- Multivariable Calculus¶

Partial Derivatives and the Jacobian - [ ] Define partial derivative \(\partial f/\partial x_i\) as the directional derivative along the standard basis vector \(e_i\) - [ ] Define the gradient \(\nabla f = (\partial f/\partial x_1, \ldots, \partial f/\partial x_n)\) as the vector of all partial derivatives - [ ] Define the directional derivative \(D_v f = \lim_{t \to 0} [f(x + tv) - f(x)]/t\). Prove \(D_v f = \nabla f \cdot v\) when \(f\) is differentiable. This is the chain rule in disguise. - [ ] Define the Jacobian matrix \(Df|_x \in \mathbb{R}^{m \times n}\) for a map \(f: \mathbb{R}^n \to \mathbb{R}^m\) as the matrix of all partial derivatives \([Df]_{ij} = \partial f_i/\partial x_j\). \(Df|_x\) is the best linear approximation to \(f\) near \(x\). - [ ] State the chain rule for compositions: if \(h = f \circ g\), then \(Dh|_x = Df|_{g(x)} \cdot\)

Change of Variables in Integrals - [ ] State the substitution formula for a change of variables \(\varphi: U \subset \mathbb{R}^2 \to V \subset \mathbb{R}^2\): \(\iint_V f(x,y)\, dx\, dy = \iint_U f(\varphi(u,v)) |\det D\varphi(u,v)|\, du\, dv\) - [ ] Compute this explicitly for polar coordinates: \(\varphi(r,\theta) = (r \cos \theta, r \sin \theta)\). Show \(|\det D\varphi| = r\). Verify the formula for the area of a disk. - [ ] Key note for later: the area element on a surface will look like \(|\varphi_u \times \varphi_v|\, du\, dv\), a generalization of this determinant formula.

Second Derivatives - [ ] Define the Hessian \(Hf\) as the matrix of second partial derivatives: \([Hf]_{ij} = \partial^2 f/\partial x_i \partial x_j\) - [ ] State Clairaut's theorem: mixed partials commute if \(f\) is \(C^2\), i.e., \(\partial^2 f/\partial x_i \partial x_j = \partial^2 f/\partial x_j \partial x_i\), so \(Hf\) is symmetric - [ ] Write the second-order Taylor expansion: \(f(x+h) \approx f(x) + \nabla f \cdot h + \tfrac{1}{2} h^\top Hf\, h + O(|h|^3)\) - [ ] Define the Laplacian \(\Delta f = \nabla^2 f = \sum_i \partial^2 f/\partial x_i^2 = \operatorname{trace}(Hf)\). Compute \(\Delta f\) for \(f(x,y) = x^2 + y^2\), \(f(x,y,z) = 1/|x|\).

Section 2: Parametric Curves in \(\mathbb{R}^3\)¶

Parametrization and Velocity - [ ] Define a parametric curve \(\gamma: I \subset \mathbb{R} \to \mathbb{R}^3\) as a smooth map. The image \(\gamma(I) \subset \mathbb{R}^3\) is the curve. - [ ] Define the velocity vector \(\gamma'(t) = d\gamma/dt\). Interpret geometrically as the tangent direction to the curve at \(\gamma(t)\). Its magnitude \(|\gamma'(t)|\) is the speed. - [ ] Define a regular curve: \(\gamma'(t) \neq 0\) for all \(t\). Explain why this is needed (without it, the parametrization can have cusps or the tangent direction is undefined). - [ ] Define arc length: \(L(\gamma) = \int_a^b |\gamma'(t)|\, dt\). Show this is independent of reparametrization (compute for \(\gamma(t)\) and \(\gamma(\varphi(s))\) where \(\varphi\) is a reparametrization). - [ ] Define arc-length parametrization (unit speed): \(|\gamma'(t)| = 1\) for all \(t\). Show every regular curve can be reparametrized by arc length.

Curvature of a Curve - [ ] Define the unit tangent vector \(T(t) = \gamma'(t)/|\gamma'(t)|\) - [ ] Define curvature \(\kappa = |dT/ds|\) where \(s\) is arc length. Interpret as how fast the tangent direction turns per unit length traveled - [ ] Compute curvature for a circle of radius \(R\): show \(\kappa = 1/R\) (larger circle \(\to\) smaller curvature \(\to\) "flatter"). This is the prototype intuition for surface curvature. - [ ] Define the principal normal vector \(N = (dT/ds)/\kappa\). It points toward the center of curvature. - [ ] State the Frenet-Serret formula: \(dT/ds = \kappa N\). This is a preview of the covariant derivative. - [ ] Key note for later: The curvature of the surface will constrain how Brownian paths can behave. On \(S^2\), each geodesic corresponds to a great circle with the same curvature as a circle of radius \(R\).

Section 3: Regular Surfaces in \(\mathbb{R}^3\)¶

Parametrized Surfaces - [ ] Define a parametrized surface chart (less formally): \(\varphi: U \subset \mathbb{R}^2 \to \mathbb{R}^3\), a smooth map. - [ ] Define the partial derivatives (tangent vectors): \(\varphi_u = \partial\varphi/\partial u\), \(\varphi_v = \partial\varphi/\partial v\). These are vectors in \(\mathbb{R}^3\). - [ ] Interpret \(\varphi_u\) and \(\varphi_v\): they span the tangent plane at \(\varphi(u,v)\). - [ ] Define a regular point: a point where \(\varphi_u \times \varphi_v \neq 0\). The cross product being nonzero means the tangent vectors are linearly independent s.t. the tangent plane is well-defined. - [ ] Define a regular surface: a surface that can be covered by patches that are all regular everywhere.

Concrete Computations: Sphere \(S^2\) - [ ] Parametrize \(S^2 = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}\) using spherical coordinates: \(\varphi(\theta,\varphi) = (\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta)\) for \(\theta \in (0,\pi)\), \(\varphi \in (0,2\pi)\) - [ ] Compute \(\varphi_\theta\), \(\varphi_\varphi\), and compute \(\varphi_\theta \times \varphi_\varphi\). Show the vector product is nonzero for \(\theta \in (0,\pi)\) (i.e., away from poles). Conclude \(S^2\) is a regular surface.

Concrete Computations: Torus \(T^2\) - [ ] Parametrize \(T^2\) with parameters \((u,v)\), \(u \in [0,2\pi)\), \(v \in [0,2\pi)\), with \(R\) = major radius (distance from z-axis to tube center), \(r\) = minor radius (tube radius): \(\varphi(u,v) = ((R + r \cos v) \cos u, (R + r \cos v) \sin u, r \sin v)\) - [ ] Compute \(\varphi_u\), \(\varphi_v\), and compute \(\varphi_u \times \varphi_v\). Show the vector product's magnitude is \(r(R + r \cos v)\). Oberserve this is not constant. The outer equator (\(v=0\)) has magnitude \(r(R+r)\), the inner equator (\(v=\pi\)) has magnitude \(r(R-r)\). - [ ] Show \(T^2\) is a regular surface (assuming \(R > r\)).

The Tangent Plane - [ ] Define the tangent plane at a point \(p = \varphi(u_0, v_0)\): \(T_p S = \operatorname{span}\{\varphi_u(u_0,v_0), \varphi_v(u_0,v_0)\} \subset \mathbb{R}^3\) - [ ] Note: the tangent plane is a 2D linear subspace of \(\mathbb{R}^3\) (technically an affine subspace of \(\mathbb{R}^3\), centered at \(p\)) - [ ] Define the unit normal \(N = (\varphi_u \times \varphi_v)/|\varphi_u \times \varphi_v|\). Show \(N\) is perpendicular to both tangent vectors. - [ ] Define the tangent space \(T_p S = \{v \in \mathbb{R}^3 : v \cdot N(p) = 0\}\). This is the set of all vectors tangent to the surface at \(p\), and a 2-dimensional subspace of \(\mathbb{R}^3\). - [ ] Note: The projection formula \(P_x v = v - (v \cdot N(x)) N(x)\) is exactly the orthogonal projection from \(\mathbb{R}^3\) onto \(T_x S\). For \(S^2\), \(N(x) = x\), recovering \(P_x v = v - (v \cdot x)x\).

Checkpoint: Before proceeding, you should be comfortable with these topics - [ ] Write the parametrization of \(S^2\) and \(T^2\). Compute \(\varphi_u\), \(\varphi_v\) for each. - [ ] Compute the unit normal \(N(x)\) for the sphere at an arbitrary point. Verify \(N(x) = x\). - [ ] Write the projection formula \(T_x S^2 \to T_x S^2\) and explain what it does geometrically. - [ ] Why is the tangent plane only defined at regular points?

Part 2: The First Fundamental Form -- Measuring Geometry on a Surface¶

Goal: Understand how to measure lengths, angles, and areas intrinsically on a surface.

Section 4: The First Fundamental Form¶

Definition and Motivation - [ ] Motivate: on a surface, how do we measure the length of a curve \(\gamma(t) = \varphi(u(t), v(t))\)? Compute \(\gamma'(t) = \varphi_u u' + \varphi_v v'\). Then \(|\gamma'|^2 = (\varphi_u u' + \varphi_v v') \cdot (\varphi_u u' + \varphi_v v') = (\varphi_u \cdot \varphi_u)(u')^2 + 2(\varphi_u \cdot \varphi_v) u'v' + (\varphi_v \cdot \varphi_v)(v')^2\) - [ ] Define the coefficients of the first fundamental form: \(E = \varphi_u \cdot \varphi_u\), \(F = \varphi_u \cdot \varphi_v\), \(G = \varphi_v \cdot \varphi_v\) - [ ] Write the metric tensor as a \(2 \times 2\) matrix: \(g = [[E, F], [F, G]]\). Conclude \(g\) is symmetric and positive definite since the surface is regular. - [ ] Rewrite arc length: \(L = \int \sqrt{E u'^2 + 2F u'v' + G v'^2}\, dt\). This is the arc length formula in terms of surface parameters. - [ ] Define the area element: \(dA = |\varphi_u \times \varphi_v|\, du\, dv = \sqrt{EG - F^2}\, du\, dv\). (Derive this: \(|\varphi_u \times \varphi_v|^2 = |\varphi_u|^2|\varphi_v|^2 - (\varphi_u \cdot \varphi_v)^2 = EG - F^2\).) - [ ] Key note for later: all of these measurements -- length, angle, area -- depend only on the functions \(E, F, G\), not on the ambient \(\mathbb{R}^3\).

Concrete Computations: \(S^2\) - [ ] With spherical parametrization \(\varphi(\theta,\varphi)\): compute \(E, F, G\) - \(E = \varphi_\theta \cdot \varphi_\theta = 1\) - \(F = \varphi_\theta \cdot \varphi_\varphi = 0\) - \(G = \varphi_\varphi \cdot \varphi_\varphi = \sin^2\theta\) - [ ] Write the metric: \(ds^2 = d\theta^2 + \sin^2\theta\, d\varphi^2\). This is the standard round metric on \(S^2\). - [ ] Compute the area element: \(dA = \sin \theta\, d\theta\, d\varphi\). Verify: \(\iint dA = 4\pi\) (total area of unit sphere). - [ ] Compute arc length of a great circle: parametrize as \(\theta \mapsto (\theta, 0)\), length \(= \int_0^\pi d\theta = \pi\). Correct.

Concrete Computations: \(T^2\) - [ ] With torus parametrization \(\varphi(u,v)\): compute \(E, F, G\) - \(E = \varphi_u \cdot \varphi_u = (R + r \cos v)^2\) - \(F = \varphi_u \cdot \varphi_v = 0\) - \(G = \varphi_v \cdot \varphi_v = r^2\) - [ ] Write the metric: \(ds^2 = (R + r \cos v)^2 du^2 + r^2 dv^2\) - [ ] Compute the area element: \(dA = r(R + r \cos v)\, du\, dv\). This is the same expression that appears in the invariant measure calculation. - [ ] Compute the total area of \(T^2\): \(\int_0^{2\pi}\int_0^{2\pi} r(R + r \cos v)\, du\, dv = 4\pi^2 Rr\)

The Metric Tensor in Index Notation - [ ] Introduce Einstein summation convention: repeated upper and lower indices are summed. Write \(g_{ij}\) for the metric tensor components (\(i,j \in \{1,2\}\)). - [ ] Define the inverse metric \(g^{ij}\): the matrix inverse of \(g_{ij}\). For the diagonal cases (\(F=0\)): \(g^{uu} = 1/E\), \(g^{vv} = 1/G\). - [ ] State: the metric \(g_{ij}\) allows us to lower indices (convert vectors to covectors). The inverse \(g^{ij}\) raises indices. This will be needed for the Laplace-Beltrami formula. - [ ] Why this notation matters: the Laplace-Beltrami operator is written \(\Delta_g f = (1/\sqrt{|g|}) \partial_i(\sqrt{|g|}\, g^{ij} \partial_j f)\).

Isometries - [ ] Define an isometry between surfaces \(S\) and \(\tilde{S}\): a diffeomorphism \(f: S \to \tilde{S}\) that preserves the first fundamental form (lengths and angles). - [ ] State: the flat torus \(\mathbb{R}^2/\mathbb{Z}^2\) is isometric to itself, but NOT isometric to the embedded torus \(T^2\) in \(\mathbb{R}^3\). They have the same topology but different geometries. - [ ] Concrete example: show that unrolling a cylinder onto a flat strip is an isometry (the metric on the cylinder is \(ds^2 = dz^2 + d\theta^2\), same as the flat metric in Cartesian coordinates after unrolling). The cylinder and the plane are locally isometric but globally different.

Section 5: The Second Fundamental Form and Curvature¶

The first fundamental form measures intrinsic geometry (lengths, angles). But how "curved" is the surface? A cylinder has \(E=G=1\), \(F=0\) just like the plane -- they are locally isometric -- yet a cylinder is clearly curved in \(\mathbb{R}^3\). The second fundamental form captures this extrinsic curvature. - [ ] Define the second derivatives of \(\varphi\): \(\varphi_{uu}, \varphi_{uv}, \varphi_{vv}\) (compute these for practice on \(S^2\) and \(T^2\)) - [ ] Note: \(\varphi_{uu}\) is not necessarily tangent to the surface. Decompose it: \(\varphi_{uu}\) = (tangential component) + (normal component). The normal component is \(LN\) where \(L = \varphi_{uu} \cdot N\). - [ ] Define the coefficients of the second fundamental form: \(L = \varphi_{uu} \cdot N\), \(M = \varphi_{uv} \cdot N\), \(N_{\text{coeff}} = \varphi_{vv} \cdot N\) (note: overloaded notation -- use e, f, g or L, M, N carefully in your notes) - [ ] Write the second fundamental form matrix: \(II = [[L, M], [M, N_{\text{coeff}}]]\)

Principal Curvatures - [ ] Define the shape operator (Weingarten map) \(W: T_pS \to T_pS\) by \(W = g^{-1} \cdot II\) (matrix product of inverse metric with second fundamental form). It measures how the normal \(N\) changes as you move along the surface. - [ ] Define principal curvatures \(k_1, k_2\): the eigenvalues of \(W\). The corresponding eigenvectors are the principal directions. - [ ] Define Gaussian curvature \(K = k_1 k_2 = \det(II)/\det(g)\). This is the product of the two principal curvatures. - [ ] Define mean curvature \(H = (k_1 + k_2)/2 = \operatorname{trace}(W)/2\). This is the average of the two principal curvatures.

Concrete Computations: \(S^2\) - [ ] Compute \(II\) for \(S^2\): show \(L = 1\), \(M = 0\), \(N_{\text{coeff}} = \sin^2\theta\) (using the computed normal \(N = -\varphi\), the inward normal for the unit sphere) - [ ] Compute \(k_1 = k_2 = 1\). \(S^2\) is a sphere with both principal curvatures equal to \(1/R\) (for unit sphere, \(R=1\)). - [ ] \(K = 1\), \(H = 1\). \(S^2\) has constant positive Gaussian curvature.

Concrete Computations: \(T^2\) - [ ] Compute \(N\) (the unit normal) for the torus: \(N(u,v) = (\cos v \cos u, \cos v \sin u, \sin v)\). Show this directly from the cross product calculation. - [ ] Compute \(L\), \(M\), \(N_{\text{coeff}}\) for the torus - [ ] Compute the principal curvatures: \(k_1 = \cos v / (R + r \cos v)\), \(k_2 = 1/r\) - [ ] Compute Gaussian curvature: \(K = \cos v / (r(R + r \cos v))\) - [ ] Key observation: \(K > 0\) when \(\cos v > 0\) (outer half of the torus), \(K = 0\) on the top/bottom circles (\(v = \pi/2, 3\pi/2\)), \(K < 0\) when \(\cos v < 0\) (inner half). The torus has regions of all three signs of curvature. Draw this. - [ ] Compute mean curvature: \(H = (R + 2r \cos v)/(2r(R + r \cos v))\)

Theorema Egregium (Statement Only) - [ ] State Gauss's Theorema Egregium: the Gaussian curvature \(K\) is an intrinsic invariant -- it can be computed from the first fundamental form alone, without reference to the ambient \(\mathbb{R}^3\). This is non-obvious: \(K = k_1 k_2\) is defined extrinsically via the shape operator, but Gauss showed it depends only on \(g_{ij}\). - [ ] State the formula (Brioschi formula): \(K =\) [expression in \(E\), \(F\), \(G\) and their derivatives -- look this up in Pressley]. You do not need to derive it, but you should state it. - [ ] Corollary: a flat map of the sphere cannot preserve distances (any map from \(S^2\) to \(\mathbb{R}^2\) must distort lengths). This is why no world map is perfectly accurate. - [ ] Project connection: Gaussian curvature will appear in the Gauss-Bonnet theorem and influences the spectral theory of the Laplace-Beltrami operator.

Checkpoint before proceeding - [ ] State the first and second fundamental forms for \(S^2\) and \(T^2\). What do their components measure? - [ ] State the Gaussian curvature of \(S^2\). Explain in words: what does \(K=1\) mean geometrically? - [ ] For the torus, identify the regions of positive, zero, and negative curvature. Draw it. - [ ] Explain in one sentence why the cylinder and the plane are locally isometric but \(S^2\) and the plane are not.

Paper 3: Geodesics and the Covariant Derivative¶

Goal: Understand geodesics (straight paths on a surface) and the covariant derivative (the notion of differentiation on a surface). These two concepts are the geometric foundation of BM on manifolds.

Section 6: Geodesics¶

On \(\mathbb{R}^n\), straight lines minimize length between two points. On a surface, what plays the role of a straight line? Two equivalent answers: (1) curves that locally minimize length, (2) curves whose acceleration is always normal to the surface (no tangential acceleration). - [ ] Define geodesic via the second characterization: \(\gamma\) is a geodesic if \(\gamma''(t)\) is perpendicular to \(T_{\gamma(t)}S\) at every point, i.e., the tangential component of \(\gamma''\) is zero. - [ ] Derive the geodesic equations in terms of surface parameters \((u(t), v(t))\): - \(u'' + \Gamma^u_{uu}(u')^2 + 2\Gamma^u_{uv}u'v' + \Gamma^u_{vv}(v')^2 = 0\) - \(v'' + \Gamma^v_{uu}(u')^2 + 2\Gamma^v_{uv}u'v' + \Gamma^v_{vv}(v')^2 = 0\) - [ ] The symbols \(\Gamma^k_{ij}\) are called Christoffel symbol. Note here that geodesics depend on the first fundamental form alone.

Christoffel Symbols - [ ] Define the Christoffel symbols of the first kind: \([ij, k] = \tfrac{1}{2}(\partial_i g_{jk} + \partial_j g_{ik} - \partial_k g_{ij})\) - [ ] Define the Christoffel symbols of the second kind: \(\Gamma^k_{ij} = g^{kl} [ij, l]\) (raise the last index with the inverse metric) - [ ] Physical interpretation: \(\Gamma^k_{ij}\) measures how much the basis vector \(\partial_j\) "tilts" in the \(e_k\) direction as you move in the \(e_i\) direction. They encode all the information about how the coordinate system curves. - [ ] Compute Christoffel symbols for \(S^2\) (spherical coordinates): - The nonzero ones are \(\Gamma^\theta_{\varphi\varphi} = -\sin \theta \cos \theta\) and \(\Gamma^\varphi_{\theta\varphi} = \Gamma^\varphi_{\varphi\theta} = \cos \theta / \sin \theta\). Verify this computation step by step - [ ] Compute Christoffel symbols for \(T^2\) (write out all nonzero ones): - \(\Gamma^u_{uv} = \Gamma^u_{vu} = -r \sin v / (R + r \cos v)\), \(\Gamma^v_{uu} = \sin v (R + r \cos v) / r\)

Geodesics on Specific Surfaces - [ ] Show that great circles satisfy the geodesic equations. Parametrize a great circle and verify. - [ ] The geodesics of the embedded torus are complicated. State that they exist (by the existence theorem for ODEs applied to the geodesic equations) but do not have simple closed-form expressions in general. - [ ] Show that Christoffel symbols vanish in Cartesian coordinates, and the geodesic equations reduce to \(u'' = v'' = 0\), so geodesics are straight lines. Verify this is consistent with the definition.

The Exponential Map - [ ] Define \(\exp_p: T_pS \to S\) by \(\exp_p(v) = \gamma(1)\) where \(\gamma\) is the geodesic starting at \(p\) with velocity \(v\) - [ ] State: \(\exp_p\) is defined on a neighborhood of 0 in \(T_pS\) and is a local diffeomorphism near 0 (by the inverse function theorem and existence/uniqueness of ODE) - [ ] Interpret geometrically: \(\exp_p(v)\) is the point you reach if you walk in direction \(v\) for \(|v|\) units of arc length along the geodesic - [ ] Key note for later: The formulation of Euler-Maruyama on a manifold uses the exponential map: \(X_{n+1} = \exp_{X_n}(\sqrt{\Delta t} \cdot Z_n)\) where \(Z_n\) is a tangent vector at \(X_n\). The projection scheme we use is an approximation to this that works well for small \(\Delta t\).

Section 7: The Covariant Derivative¶

If \(V(t)\) is a vector field along a curve \(\gamma(t)\) on a surface, and we want to differentiate \(V\), the naive derivative \(V'(t) = dV/dt\) is a vector in \(\mathbb{R}^3\) that may point off the surface. We need a notion of derivative that stays tangential. - [ ] Concrete example: take \(\gamma(t)\) to be a meridian on \(S^2\) (a great circle at fixed \(\varphi\)). The tangent field \(V(t) = \gamma'(t)\) is always tangent to \(S^2\). Compute \(V'(t)\) in \(\mathbb{R}^3\) -- it points toward the center of the sphere, normal to \(S^2\). So \(V'(t)\) is not a tangent vector. The "intrinsic" rate of change of \(V\) along \(\gamma\) is zero (\(V\) is a geodesic, so it parallel transports its own tangent vector), but the ambient derivative is nonzero. - [ ] Define the covariant derivative: \(\nabla_{\gamma'(t)} V = (V'(t))_T = V'(t) - (V'(t) \cdot N) N\), the projection of \(V'(t)\) onto the tangent plane. This is the tangential component of \(V'(t)\).

Parallel Transport - [ ] State that a vector field \(V(t)\) along \(\gamma(t)\) is parallel if \(\nabla_{\gamma'(t)} V = 0\), i.e., the covariant derivative is identically zero along the curve. The vector is "not rotating intrinsically as you move along \(\gamma\). - [ ] An example: transport a tangent vector on \(S^2\) along a spherical triangle (\(1/8\) of the sphere). When you return to the start, the vector has rotated by \(\pi/2\). - [ ] State parallel transport preserves the inner product between vectors. If \(V, W\) are both parallel along \(\gamma\), then \(d/dt (V \cdot W) = 0\). - [ ] Key note for later: The parallel transport equation in coordinates is \(dV^k/dt + \Gamma^k_{ij} (dx^i/dt) V^j = 0\). The Christoffel symbols measure the "correction" needed to keep a vector parallel.

The Covariant Derivative for Vector Fields - [ ] Define the Levi-Civita connection \(\nabla_X Y\) for vector fields \(X, Y\) on \(S\): it is the unique connection that (1) is compatible with the metric (preserves inner products under parallel transport) and (2) is torsion-free (\(\nabla_X Y - \nabla_Y X = [X,Y]\)). These two conditions uniquely determine all the Christoffel symbols \(\Gamma^k_{ij}\). - [ ] In coordinates, \(\nabla_{\partial_i} \partial_j = \sum_k \Gamma^k_{ij} \partial_k\). The Christoffel symbols are exactly the components of the covariant derivative of basis vectors. - [ ] Project connection: The covariant derivative is why BM on a manifold requires the Stratonovich formulation. The Itô formula with ordinary derivatives does not transform correctly under isometries. The Stratonovich formula uses the Levi-Civita connection so it does transform correctly.

The Riemann Curvature Tensor - [ ] Define the Riemann curvature tensor as \(R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z\). It measures the failure of covariant derivatives to commute. - [ ] Note that for surfaces in \(\mathbb{R}^3\), \(R\) is determined entirely by the Gaussian curvature \(K\). - [ ] Note that on a flat plane, parallel transport around any closed loop returns the vector unchanged (\(R=0\)). On a curved surface, parallel transport around a loop rotates the vector. The rotation angle equals \(K\) times the enclosed area (Gauss-Bonnet). This is why holonomy measures curvature.

Checkpoint before proceeding - [ ] Define the covariant derivative \(\nabla_{\gamma'} V\) in words and in formula - [ ] Explain the parallel transport example on \(S^2\) (spherical triangle). Why does the vector rotate? - [ ] Write the geodesic equation using covariant derivatives: \(\nabla_{\gamma'} \gamma' = 0\) (the acceleration is zero intrinsically) - [ ] What are Christoffel symbols measuring? Write the formula for \(\Gamma^k_{ij}\) in terms of the metric. - [ ] Why does the covariant derivative matter for stochastic processes on manifolds?

Paper 4: Abstract Manifolds, the Laplace-Beltrami Operator¶

Goal: Generalize from surfaces in \(\mathbb{R}^3\) to abstract Riemannian manifolds. Define the Laplace-Beltrami operator and compute it explicitly on \(S^2\) and \(T^2\).

Section 8: Smooth Manifolds¶

The Poincaré hyperbolic disk \(H^2\) is not naturally a surface in \(\mathbb{R}^3\) with an induced metric. It is defined abstractly as a set with a Riemannian metric imposed on it. To handle \(H^2\), we need a more absract framework. For \(S^2\) and \(T^2\), this framework should produce the same results as the embedded form -- so mastering the embedded case first is the right approach.

[ ] Define a topological manifold of dimension \(n\): a Hausdorff topological space \(M\) such that every point has a neighborhood homeomorphic to \(\mathbb{R}^n\)
[ ] Define a chart (more formally) \((U, \varphi)\): an open set \(U \subset M\) and a homeomorphism \(\varphi: U \to \mathbb{R}^n\). The functions \(\varphi\) give local coordinates.
[ ] Define a smooth atlas: a collection of charts \(\{(U_\alpha, \varphi_\alpha)\}\) that cover \(M\), such that all transition maps \(\varphi_\beta \circ \varphi_\alpha^{-1}\) are smooth
[ ] Define a smooth manifold: a topological manifold with a smooth atlas
[ ] Define the tangent space \(T_pM\) at \(p \in M\): the set of equivalence classes of curves through \(p\), where two curves are equivalent if they have the same velocity in local coordinates. This abstracts the tangent plane.
[ ] State: \(T_pM\) is an \(n\)-dimensional real vector space. The tangent bundle \(TM = \cup_p T_pM\) is the collection of all tangent spaces.

Riemannian Metric - [ ] Define a Riemannian metric \(g\) as an assignment of an inner product \(g_p: T_pM \times T_pM \to \mathbb{R}\) to each tangent space \(T_pM\), varying smoothly in \(p\) - [ ] State in local coordinates \((x^1, \ldots, x^n)\), \(g\) is a smooth matrix-valued function \(g_{ij}(x)\) that is symmetric and positive definite at every point - [ ] Define a Riemannian manifold as a smooth manifold with a Riemannian metric - [ ] For a surface \(S \subset \mathbb{R}^3\) with parametrization \(\varphi\), the first fundamental form \(g_{ij} = \partial_i\varphi \cdot \partial_j\varphi\) is exactly a Riemannian metric in the abstract sense. The abstract framework recovers everything we have done.

The Poincaré Disk \(H^2\) - [ ] Define \(H^2 = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 < 1\}\) (the open unit disk) with metric: \(g_{ij} = (4/(1-r^2)^2) \delta_{ij}\) where \(r^2 = x^2 + y^2\) - [ ] This means \(ds^2 = 4(dx^2 + dy^2)/(1-r^2)^2\) -- the metric is the Euclidean metric scaled by a position-dependent conformal factor - [ ] Note: as \(r \to 1\) (approaching the boundary), the conformal factor blows up; distances near the boundary are much larger than they appear in the disk. This is why "equivalent" triangles near the boundary look smaller. - [ ] Compute the area element: \(dA = 4\, dx\, dy / (1-r^2)^2\). Show the total area of \(H^2\) is infinite. - [ ] State the Gaussian curvature of \(H^2\): \(K = -1\) everywhere. \(H^2\) is the unique (up to scaling) simply connected surface of constant negative curvature. - [ ] Project connection: BM on \(H^2\) does not converge to a uniform distribution as \(H^2\) has infinite volume. Instead, BM paths converge to the boundary circle almost surel.y

Section 9: The Laplace-Beltrami Operator¶

Motivation On \(\mathbb{R}^n\), the Laplacian \(\Delta f = \sum \partial^2 f/\partial x_i^2\) generates Brownian motion. If we simulate BM in \(\mathbb{R}^n\) and test it with a smooth function \(f\), the expected rate of change of \(f\) along the path is \(\tfrac{1}{2}\Delta f\). We need an analog of this fact for a Riemannian manifold.

Definition - [ ] The gradient in \(\mathbb{R}^n\) satisfies \(\nabla f \cdot v = D_v f\) (directional derivative). On a manifold, define the Riemannian gradient \(\operatorname{grad} f\) as the unique tangent vector satisfying \(g(\operatorname{grad} f, v) = df(v)\) for all tangent vectors \(v\). - [ ] In coordinates, \((\operatorname{grad} f)^i = g^{ij} \partial_j f\).= - [ ] Define the divergence of a vector field \(X\): \(\operatorname{div} X = (1/\sqrt{|g|}) \partial_i(\sqrt{|g|}\, X^i)\). This is the generalization of \(\nabla \cdot X\) that accounts for the metric's volume distortion. - [ ] Define the Laplace-Beltrami operator as \(\Delta_g f = \operatorname{div}(\operatorname{grad} f) = (1/\sqrt{|g|}) \partial_i(\sqrt{|g|}\, g^{ij} \partial_j f)\). Show this reduces to the ordinary Laplacian in Cartesian coordinates on \(\mathbb{R}^n\): \(g_{ij} = \delta_{ij}\), \(\sqrt{|g|} = 1\), so \(\Delta_g f = \partial_i(\partial^i f) = \sum \partial^2 f/\partial x_i^2\).

Concrete Computations: \(S^2\) - [ ] With metric \(g_{\theta\theta} = 1\), \(g_{\varphi\varphi} = \sin^2\theta\), \(F=0\): compute \(\sqrt{|g|} = \sin \theta\) - [ ] Compute \(\operatorname{grad} f\): \((\operatorname{grad} f)^\theta = \partial f/\partial\theta\), \((\operatorname{grad} f)^\varphi = (1/\sin^2\theta) \partial f/\partial\varphi\) - [ ] Compute \(\Delta_{S^2} f = (1/\sin \theta) [\partial/\partial\theta(\sin \theta\, \partial f/\partial\theta) + (1/\sin \theta) \partial^2 f/\partial\varphi^2]\) - [ ] Verify for \(f(\theta,\varphi) = \cos \theta\) (which is \(z\) restricted to \(S^2\)), compute \(\Delta_{S^2} f = -2\cos \theta = -2f\). Conclude that \(f = \cos \theta\) is an eigenfunction of \(\Delta_{S^2}\) with eigenvalue \(-2 = -l(l+1)\) for \(l=1\). - [ ] State the eigenfunctions of \(-\Delta_{S^2}\) are the spherical harmonics \(Y^m_l\) with eigenvalues \(l(l+1)\). They form a complete orthonormal basis for \(L^2(S^2)\). This is the spectral theory of the sphere.

Concrete Computations: \(T^2\) - [ ] With metric \(g_{uu} = (R + r \cos v)^2\), \(g_{vv} = r^2\), \(F=0\): compute \(\sqrt{|g|} = r(R + r \cos v)\) - [ ] Compute \(\Delta_{T^2} f = [1/(r(R + r \cos v))] \{ \partial/\partial u[(r/(R + r \cos v)) \partial f/\partial u] + \partial/\partial v[((R + r \cos v)/r) \partial f/\partial v] \}\) - [ ] Simplify for u-direction: since \(\partial/\partial u[(1/(R+r \cos v)) \partial f/\partial u] = (1/(R+r \cos v)) \partial^2 f/\partial u^2\) - [ ] Simplify for v-direction: this involves a derivative of \((R + r \cos v)\) which produces a \(\sin v\) term - [ ] Write the full expression. Note it is not separable (the v-coefficient of \(\partial^2 f/\partial v^2\) depends on \(v\)), confirming that the embedded torus does not have a clean eigenfunction decomposition.

The Invariant Measure Connection - [ ] State: the Laplace-Beltrami operator is self-adjoint with respect to the Riemannian volume measure \(d\mu_g = \sqrt{|g|}\, du\, dv\). This means \(\int f \Delta_g h\, d\mu_g = \int (\Delta_g f) h\, d\mu_g\) for smooth \(f, h\). - [ ] State and derive: the Riemannian volume measure \(d\mu_g\) is the invariant measure of BM. That is, if \(X_0\) has distribution \(d\mu_g\) (normalized), then \(X_t\) has the same distribution for all t. This follows from the self-adjointness of \(\Delta_g\). - [ ] For \(T^2\): \(d\mu_g = r(R + r \cos v)\, du\, dv\). This is not uniform. Write out the normalized version: the probability density is \(p(u,v) = r(R + r \cos v) / (4\pi^2 Rr) = (R + r \cos v)/(4\pi^2 R)\). This is the theoretical target distribution your simulation must match. - [ ] For \(S^2\): \(d\mu_g = \sin \theta\, d\theta\, d\varphi\). Normalized: \(p(\theta,\varphi) = \sin \theta / (4\pi)\). Uniform in \(\varphi\), biased toward the equator (where \(\sin \theta\) is larger). In Cartesian coordinates this is just uniform on the sphere.

Checkpoint before proceeding - [ ] Write the formula for \(\Delta_g f\) from memory. Define every symbol in it. - [ ] Compute \(\Delta_{S^2}(\cos \theta)\) and confirm it equals \(-2\cos \theta\) - [ ] What is the invariant measure for BM on the embedded \(T^2\)? Why is it not uniform in \((u,v)\)? - [ ] State the relationship between \(\Delta_g\) and the generator of BM on a manifold

Paper 5: Stochastic Processes on Riemannian Manifolds¶

Goal: Connect everything above to our proposed simulation. Understand why the projection scheme works, why Stratonovich is required, and what "Brownian motion on a manifold" means precisely.

Section 10: Itô vs. Stratonovich -- The Coordinate Problem¶

Review of Itô Calculus on \(\mathbb{R}^n\) - [ ] State Itô's formula: for \(X_t\) satisfying \(dX = b\, dt + \sigma\, dW\) and \(f\) smooth: \(df(X_t) = \partial f/\partial x_i\, dX^i + \tfrac{1}{2} \sigma\sigma^\top_{ij} \partial^2 f/\partial x_i \partial x_j\, dt\). The second-order term is the Itô correction. - [ ] State Stratonovich's formula: for the same \(X_t\), with \(\circ\) denoting Stratonovich integral: \(df(X_t) = \partial f/\partial x_i \circ dX^i\). No second-order correction. The chain rule holds in its classical form. - [ ] State the conversion formula: \(X \circ dW = X\, dW + \tfrac{1}{2} d[X, W]_t\) where \([X, W]_t\) is the quadratic covariation. For \(\sigma(X_t)\): Stratonovich drift = Itô drift + \(\tfrac{1}{2} \sigma(X) \sigma'(X)\).

Why Itô Fails on Manifolds - [ ] State the problem: suppose \(X_t \in S\) and we change coordinates \(y = \varphi(x)\) (a diffeomorphism). The Itô formula for \(y(X_t)\) gives: \(dy = D\varphi(X)\, dX + \tfrac{1}{2} \operatorname{trace}(D^2\varphi(X) \sigma\sigma^\top)\, dt\). The second term is NOT intrinsic -- it depends on the coordinate change \(\varphi\), not just the geometry of \(S\). - [ ] Consequence: if you write an Itô SDE in one coordinate chart and transform it to another chart, you get a different SDE with extra drift terms. This means Itô SDEs are NOT coordinate-invariant -- two observers using different parametrizations would disagree about what process is running. - [ ] State: the Stratonovich SDE transforms correctly: \(dy = D\varphi(X) \circ dX\), with no extra drift terms. Stratonovich SDEs are coordinate-invariant. - [ ] Conclusion: the intrinsic Brownian motion on a manifold must be defined via a Stratonovich SDE. Any Itô representation will have extra drift terms (the Itô-Stratonovich correction) involving the Christoffel symbols.

The Itô-Stratonovich Correction on \(S^2\) - [ ] State the Itô form of BM on \(S^2\) (unit sphere in \(\mathbb{R}^3\)): \(dX^i = (P_{ij}(X)/1) dW^j - \tfrac{1}{2}(n-1) X^i\, dt\) where \(P_{ij}(x) = \delta_{ij} - x_i x_j\) is the projection matrix and \(n=3\) - [ ] Interpret: the drift term \(-X^i\, dt\) points inward (toward the origin). In the Itô representation, BM on \(S^2\) needs a drift to stay on the sphere. - [ ] The Stratonovich form: \(dX^i = P_{ij}(X) \circ dW^j\). No drift term. The projection handles everything. - [ ] Verify the two are equivalent by computing the Itô correction: for \(\sigma_{ij} = P_{ij}(X)\), \(\tfrac{1}{2} \sum_j (\partial\sigma_{ij}/\partial x_k) \sigma_{kj} = -X^i\). So Stratonovich = Itô form + drift \(-X^i\, dt\). The Itô-to-Stratonovich correction exactly cancels the inward drift.

Why the Projection Scheme Implements Stratonovich - [ ] The Euler-Maruyama scheme for the Stratonovich SDE \(dX = \sigma(X) \circ dW\): \(X_{n+1} = X_n + \sigma(X_n) \Delta W_n + \tfrac{1}{2} \sum_j (\partial\sigma_{ij}/\partial x_k\, \sigma_{kj})(X_n) \Delta t + O(\Delta t^{3/2})\) - [ ] Understand how projection is used: \(\tilde{X}_{n+1} = X_n + \sqrt{\Delta t} \cdot P_{X_n}(Z_n)\), then \(X_{n+1} = \tilde{X}_{n+1}/|\tilde{X}_{n+1}|\) (for sphere) - [ ] Show: to first order in \(\Delta t\), \(X_{n+1} = X_n + P_{X_n}(\sqrt{\Delta t}\, Z_n) - \tfrac{1}{2} (X_n \cdot (\sqrt{\Delta t}\, Z_n)^2) X_n + O(\Delta t^{3/2})\). The second term contributes an \(O(\Delta t)\) drift. Compute its expectation: \(E[\tfrac{1}{2} (X_n \cdot \sqrt{\Delta t}\, Z_n)^2 X_n] = \tfrac{1}{2} \Delta t \cdot E[Z^T P_X Z] X_n = \tfrac{1}{2} \Delta t (n-1) X_n\, dt\) where \(n-1 = 2\) for \(S^2 \subset \mathbb{R}^3\). This is exactly the Itô-Stratonovich correction, so the scheme is consistent with the Stratonovich formulation. - [ ] Conclusion to write in notes: the re-normalization after each step automatically incorporates the Itô correction, so the projection scheme implements Stratonovich EM without any extra drift term needing to be computed explicitly.

Section 11: Brownian Motion on a Manifold¶

Definition - [ ] Define Brownian motion on a Riemannian manifold \((M, g)\): a continuous stochastic process \(X_t\) on \(M\) with generator \(\tfrac{1}{2}\Delta_g\). Equivalently, for every smooth \(f: M \to \mathbb{R}\), \(f(X_t) - f(X_0) - \tfrac{1}{2}\int_0^t \Delta_g f(X_s)\, ds\) is a martingale. - [ ] State (without proof): BM on \(M\) can be constructed as the solution to the Stratonovich SDE on \(M\) corresponding to choosing \(\sigma\) to be the "square root" of the metric \(g\). For \(S^2\) and \(T^2\), the projection scheme implements this. - [ ] State: BM on \(M\) is the diffusion whose transition density \(p_t(x, y)\) solves the heat equation \(\partial_t p = \tfrac{1}{2}\Delta_g p\) with initial condition \(p_0(x, \cdot) = \delta_x\).

The Heat Kernel - [ ] Define the heat kernel \(p_t(x, y)\): the fundamental solution to \(\partial_t u = \tfrac{1}{2}\Delta_g u\). It is simultaneously (1) the transition density of BM (\(P_x(X_t \in dy) = p_t(x,y)\, d\mu_g(y)\)), and (2) the Green's function for the heat equation. - [ ] State properties: \(p_t(x,y) > 0\), \(\int p_t(x,y)\, d\mu_g(y) = 1\), \(p_{s+t}(x,y) = \int p_s(x,z) p_t(z,y)\, d\mu_g(z)\) (Chapman-Kolmogorov) - [ ] Heat kernel on \(S^2\): via spectral expansion. If \(\varphi_l\) are eigenfunctions of \(\Delta_g\) with eigenvalues \(-\lambda_l\): \(p_t(x,y) = \sum_l e^{-\lambda_l t/2} \sum_m \varphi_{lm}(x) \varphi_{lm}(y)\). For \(S^2\), \(\lambda_l = l(l+1)\), eigenfunctions are spherical harmonics \(Y^m_l\), and: \(p_t(x,y) = \sum_{l=0}^\infty [(2l+1)/(4\pi)] e^{-l(l+1)t/2} P_l(\cos d(x,y))\) where \(d(x,y)\) is geodesic distance and \(P_l\) is the Legendre polynomial. - [ ] Verify: as \(t \to \infty\), all terms except \(l=0\) vanish, leaving \(p_\infty(x,y) = 1/(4\pi)\) = uniform density on \(S^2\). This is the correct long-time behavior on \(S^2\). - [ ] For \(T^2\): the spectral expansion does not simplify cleanly (see Paper 4, Section 9 notes). The long-time limit is \(p_\infty(u,v) = (R + r \cos v)/(4\pi^2 R)\), which is the normalized invariant measure.

Euler-Maruyama Convergence - [ ] State the strong convergence theorem for EM: under Lipschitz conditions on \(\sigma\) and \(b\), the EM scheme converges with strong order \(\tfrac{1}{2}\) (i.e., \(E[|X_T - \tilde{X}_T|^2] = O(\Delta t)\)). - [ ] State the weak convergence theorem: for smooth test functions \(f\), \(|E[f(X_T)] - E[f(\tilde{X}_T)]| = O(\Delta t)\). Weak order 1. - [ ] State: the geometric projection introduces an \(O(\Delta t)\) error per step from the constraint manifold (points are pushed slightly off the surface before renormalization). This does not degrade the order of convergence but affects the constant. - [ ] Practical implication: use \(\Delta t \leq 0.01\) for \(S^2\) and \(T^2\) simulations. At \(\Delta t = 0.1\), the approximation error is large enough to see visually in the distribution plots.

Checkpoint - [ ] Write the Stratonovich SDE for BM on \(S^2\) - [ ] Explain why Itô fails on manifolds (coordinate non-invariance argument) - [ ] Derive the Itô form of BM on \(S^2\) and identify the inward drift term - [ ] Explain why the projection + renormalize scheme implements Stratonovich and not Itô - [ ] Write the heat kernel formula for \(S^2\); identify the eigenvalues, eigenfunctions, and long-time limit - [ ] State the invariant measures for \(S^2\) and embedded \(T^2\) and explain where they come from