Differential Geometry

• Parametrization
• A vector-valued continuous function \(\gamma\colon I\to \mathbb{R}^n\) form an interval \(I\) to a Euclidean space.

• Two parametric \(C^k\)-curves \(\gamma_1\colon I_1 \to \mathbb{R}^n, \gamma_2\colon I_2 \to \mathbb{R}^n\) are said to be equivalent if there exists a bijective \(C^r\)-map \(\varphi\colon I_1 \to I_2\) such that \[ \forall t\in I_1, \varphi'(t) \neq 0, \gamma_2\circ \varphi = \gamma_2 \]
• \(\gamma_2\) is said to be a re-parametrization of \(\gamma_1\).
• Notice that the same subset of \(\mathbb{R}^n\) can be parameterized in multiple non-equivalent ways, with varying self-crossing and differentiablity.

• The equivalence classes are called \(C^k\)-curves.
• The equivalence class may be alternatively called the \(C^k\)-arc, and the parametrization is called the \(C^k\)-curve.

• The equivalence classes under the equivalence relation with additional requirement \(\varphi(t) >0\).

• A curve \(C\) is differentiable if there exists a differentiable parametrization \(\gamma\).

• A curve \(C\) is \(C^{k}\) if there exists a \(C^k\)-parametrization \(\gamma\).

• A curve \(C\) is regular if there exists a \(C^1\)-parametrization \(\gamma\) with \(\forall t,\gamma'(t) \neq 0\).
• \(C^1\)-curve may have cusps, for the case of the pathological equivalence class that is other than the usual one although describing the same curve. The equivalence class that is regular will not have such pathological cases.
• It would coincide with the constructive definition of regular curve? at least for the regular of order 1?
• Theorem
  • Regular curves does not have cusps
  • For a regular curve \(\gamma\colon I_1\to \mathbb{R}^2\),
  • At every point \(t_0\in I_1\), there is a reparametrization \(\alpha\colon I_2\to \mathbb{R}^2\) and a Cartesian coordinate system \(\bar{x}, \bar{y}\colon \mathbb{R}^2\to \mathbb{R}\) in which \(\alpha(t)\) is identical to \((t,f(t))\) for a locally \(C^1\)-function \(f\)

• A curve \(C\) is smooth regular if there exists a parametrization that is both \(C^r\) and regular.

• A curve \(C\) is closed if the infinite concatenation of any parametrizations with itself is periodic.

• A curve that does not intersect itself.
• A curve \(C\) is simple if there exists an injective parametrization.

• A curve \(C\) is simple closed if there exists an injective parametrization except at one point, and its periodic when infinitely concatenated.
• Theorem (Jordan Curve)
  • Any Jordan (simple closed) curve separates \(\mathbb{R}^2\) in two connected components, one of them bounded and the other unbounded.

• Finite length.

• Approximation of a given curve by finite number of connected line segments.
• The arclength may not be obtained by the rectification itself, but the supremum can give the arclenth in the case of the rectifiable curves.

• Koch Curve, Koch Star, Koch Island

Example of a non-rectifiable curve.

• Bounded curve with infinite length.
• Continuous curve where tangent line at any point is indetermined.
• Koch snowflake - Wikipedia

• \[ \operatorname{length}(\gamma) := \sup_P\left\{\sum_{j=1}^k|\gamma(t_j) - \gamma(t_{j-1})|\right\} \]
• where \(P\) is the partition on the domain \([a,b]\) of the parametrization \(\gamma\).
  • A curve with finite length is a rectifiable curve.
• The reparametrization induces natural isomorphism in terms of the length between partitions of two intervals, therefore the length is invariant under reparametrization. (Property 2)

• \(\operatorname{length}(\gamma) \ge |\gamma(b) - \gamma(a)|\), with equality when \(\gamma\) is a straight line.
• For two equivalent parametrization \(\gamma_1,\gamma_2\), \(\operatorname{length}(\gamma_1) = \operatorname{length}(\gamma_2)\)
• \(\operatorname{length}(\gamma_1\ast \gamma_2) = \operatorname{length}(\gamma_1) + \operatorname{length}(\gamma_2)\) where \(\ast\) denotes the concatenation.
• The length is invariant under isometry.
• Given a sequence of curves \((\gamma_n)_{n=1}^\infty\) that converges pointwise to \(\gamma\), \[ \operatorname{length}(\gamma) \le \operatorname*{lim\,inf}_{n\to \infty}\operatorname{length}(\gamma_n) \]
• \(\operatorname{length}\) is the only function satisfying that satisfying the above five properties.
  • If a function \(\lambda\colon \mathbf{Curve} \to \mathbb{R}\cup \{\infty\}\) satisfies the above five properties, then \(\lambda = \operatorname{length}\).
• If a curve can be parameterized by a Lipschitz continuous function \(\gamma\colon [a,b] \to X\), then it is rectifiable, and the speed of \(\gamma\) at \(t\in [a,b]\) can be defined as a metric derivative: \[ \operatorname{speed}_\gamma(t) := \operatorname*{lim\,sup}_{s\to t}\frac{d(\gamma(s),\gamma(t))}{|s-t|} \] which then satisfies: \[ \operatorname{length}(\gamma) = \int_a^b\operatorname{speed}_\gamma(t)\,dt. \]

• Given a rectifiable plane curve \(\gamma\),
• \(n_\gamma(\ell)\colon \mathbf{2\text{-}OrientedLines}\to \mathbb{Z}\) which defined as
  • the number of intersections between a line \(\ell \in \mathbf{2\text{-}OrientedLines}\) and the curve \(\gamma\),
  • and can be parametrized by the signed distance from the origin \(p\) and the direction \(\varphi\) in which it points
• satisfies: \[ \operatorname{length}(\gamma) = \frac{1}{4}\iint n_\gamma(\varphi, p)\,d\varphi\,dp. \]
  • The right hand side is usually called the Favard length.
• Equivalently, using the projection \(\pi_\theta\) onto the line through the origin with angle \(\theta\): \[ \operatorname{length}(\gamma) = \frac{1}{4}\int_0^{2\pi}\operatorname{length}(\pi_\theta\circ\gamma)\,d\theta. \]

• The differential form \(d\varphi\wedge dp\), which is called the kinematic measure, is invariant under rigid motions, and therefore the integral itself, constituting a natural integration measure for an "average" number of intersections,

• In \(\mathbb{R}^n\), the integration over the space of oriented lines is: \[ \operatorname{area}(S) = \frac{1}{2|B_1^{n-1}|}\iint n_S(\varphi, p)\,d\varphi\,dp \]
• where \(S\) is the surface of codimension 1, and \(|B_1^{n-1}|\) is the \((n-1)\)-volume of the unit ball in \(\mathbb{R}^{n-1}\), and \(d\varphi\wedge, dp\) is the appropriate kinematic measure.
• The space of oriented lines in \(\mathbb{R}^n\) is the tangent bundle of \(S^{n-1}\).

• Given a parametric curve \(\gamma\colon [a,b]\to \mathbb{R}^n\), the arclength function \(s\colon [a,b] \to [0,\infty]\) is defined to be:
• \[ s(t) := \operatorname{length}\left(\gamma\big|_{[a,t]}\right). \]
• It admits the inverse function \(s^{-1}\colon [0, \operatorname{length}(\gamma)] \to [a,b]\), since it is an monotone injection.
• For a parametric \(C^1\)-curve \(\gamma\colon [a,b]\to \mathbb{R}^n\), the arclength can equivalently be obtained by \[ s(t) = \int_a^t |\gamma'(t)|\,dt. \]
  • Notice that this formula is invariant under reparametrization, since \[ \gamma'(t)\,dt = \tilde{\gamma}'(\tilde{t})\,d\tilde{t} = \tilde{\gamma}'(\varphi(t))\varphi'(t)\,dt \] and the arclength is taken to be positive. If the curve was oriented it might introduces a sign change.

• A parametrization \(\gamma\colon I\to X\) such that for any \(t_1, t_2 \in I\) with \(t_1 \le t_2\), \[ \operatorname{length}\left(\gamma\big|_{[t_1,t_2]}\right) = t_2 - t_1. \]
• Equivalently, the reparametrization of a parametrization \(\gamma\) by the arclength: \[ \gamma\circ s^{-1}(t). \]

• \(C^1\)-parametrization \(\gamma\) is the arclength parametrization if and only if \(\forall t\in [a,b], |\gamma'(t)| = 1\).
• The arclength parametrization of a smooth regular curve is smooth and regular.

• Frenet Frame, TNB Frame
• It is a reference frame that moves along a curve.

• Given a locally regular \(C^1\)-curve \(\gamma\),
• \[ \mathbf{T}(t) := \frac{\gamma'(t)}{|\gamma'(t)|} = \tilde{\gamma}'(s) \]

• \[ \mathbf{N}(s) := \frac{\tilde{\gamma}''(s)}{\kappa(s)} \]

• \[ \mathbf{B} := \mathbf{T}\times \mathbf{N} \]

• \[ -\tau\mathbf{N} := \mathbf{B}'(s) \]

• If a spherical curve \(\gamma\colon : I\to \mathbb{S}^2\) is closed and has length less than \(2\pi\), then it is strictly contained in a hemisphere.
• Corollary
  • There is \(v\in \mathbb{S}^2\) with \(\forall t, \langle v,\gamma(t)\rangle >0\)

• If \(\gamma\colon I\to \mathbb{R}^3\) is closed and piecewise smooth regular, then \(\Phi(\gamma) \ge 2\pi\).

• For a piecewise smooth regular curve \(\gamma\colon I\to \mathbb{R}^3\), \[ \Phi(\gamma) = \sup_{p_i\ \text{rectification}}\{\Phi(p_0,\dots,p_n)\} \]

• If \(\gamma\colon I\to \mathbb{R}^3\) is closed and \(\forall t, |\gamma(t)|\le 1\), then \[ \Phi(\gamma)\ge \operatorname{length}(\gamma). \]

• Crofton formula for spherical curves.
• If \(\gamma\colon I\to \mathbb{S}^2\) is piecewise smooth regular, then \[ \operatorname{length}(\gamma) = \frac{1}{4\pi}\int_{\mathbb{S}^2}\operatorname{length}(\gamma_v^*)\,dv \]
  • where \(\gamma_v^*\) is the projection of \(\gamma\) onto the unit circle perpendicular to \(v\).

• If \(\gamma\colon I\to \mathbb{R}^2\) is piecewise smooth regular and closed, \[ \Psi(\gamma) = 2\pi m, m\in \mathbb{Z} \]
  • where \(m\) being the winding numbers.
  • In particular, for the case of simple closed curve, it is either \(\pm 2\pi\).
• If \(\gamma\) is smooth, \[ \Psi(\gamma) = \theta(b) - \theta(a). \]

• Tangent curves \(\gamma_1\) and \(\gamma_2\) are co-oriented if \(\gamma_1'\cdot \gamma_2' > 0\).
• Likewise, they are counter-oriented if \(\gamma_1' \cdot \gamma_2' < 0\).

• For two tangent curves \(\gamma_1, \gamma_2\) at \(p_0\),
• \(\gamma_1\) locally supports \(\gamma_2\) at \(p_0\) if \[ \exists \varepsilon >0, \exists R\subset \mathbb{R}^2, \gamma_1[(t_1-\varepsilon, t_1+\varepsilon)] \subset \partial R\land \gamma_2[(t_2-\varepsilon, t_2+\varepsilon)] \subset R \]
• If the curves are co-oriented and \(\gamma_1\) goes counter-clockwise (or clockwise) around \(R\), then it is said that \(\gamma_1\) locally supports \(\gamma_2\) from the right (or from the left).
  • In some local coordinate system \(f_1(x) \le f_2(x)\).
  • If \(k_1(t_1) < k_2(t_2)\), then \(\gamma_1\) supports \(\gamma_2\) form the right.
• \(\gamma_1\) globally supports \(\gamma_2\) from the outside, if \(\gamma_2\) lies entirely within the interior of \(\gamma_1\).

• Co-oriented tangent circle, with the same curvature.

• For a simple smooth regular closed curve,
• If \(\forall t, \kappa(t) \le 1\), then \(\gamma\) contains a unit disk in its interior.

• For a smooth regular curve \(\gamma\colon [a,b]\to \mathbb{R}^2\),
• the point \(\gamma(t_0)\) is called a vertex if \(k'(t_0)=0\)

• A closed smooth regular curve \(\gamma\colon \mathbb{S}^1\to \mathbb{R}^2\) has at least four vertices.

• Two curves in the plane intersecting at a point \(p\) are said to have:
  • 0th-order contact if the curves have a simple crossing (not tangent).
  • 1st-order contact if the two curves are tangent.
  • 2nd-order contact if the curvatures of the curves are equal. Such curves are said to be osculating.
  • 3rd-order contact if the derivatives of the curvature are equal.
  • 4th-order contact if the second derivatives of the curvature are equal.

• Osculate, "to touch", from Latin osculum, "kiss"
• Plane curve from a given family that has the highest possible order of contact with another curve.

• The equivalence class with respect to the equivalence relation of the same contact.

• It is a point on a curve where a moving point must reverse direction.

• A function \(f\) is said to be of type \(A_k^\pm\) if it lies in the orbit of \(x^2\pm y^{k+1}\), i.e. there exists a diffeomorphic change of coordinate in source and target which takes \(f\) into one of these form.
• Equivalence class \(A_k^\pm\) of the normal form \(x^2\pm y^{k+1}\).

• Type \(A_2\)-singularity, given by \(x^2 - y^3 = 0\).

• From Greek, 'beak-like'

In particular, if the manifold is a Euclidean space \(\mathbb{R}^n\), then the existence of \(C^k\)-diffeomorphism can be reduced to the existence of coordinate systems at each point for which a neighborhood is a graph of a \(C^k\)-function.

• The neighborhood can instead be described by the smooth regular \(\varphi\colon U\to \mathbb{R}^n\), called ((66806d22-8f92-488f-a5d9-fc26c16fcb09)), from a open connected subset \(U\subset\mathbb{R}^2\).

• For an open subset \(U\subset \mathbb{R}^{m}\) and a smooth map \(\varphi\colon U\to \mathbb{R}^n\) with \(n\ge m\),
• \(\varphi\) is said to be regular if \[ \left\{\frac{\partial \varphi}{\partial x_i}\right\}_{i=1}^m \]
• is a linearly indepnedent set of vectors?

• A smooth surface \(\Sigma\) is orientable if there exists a continuous map \(N\colon\Sigma \to \mathbb{S}^2\), called orientation or Gauss map, such that \(\forall p\in \Sigma, N(p)\perp T_p\Sigma\).

• Shape operator \(S_p\colon T_p\Sigma \to T_{p}\Sigma\) is (inclusion of) the negative of the ((66806d1e-2876-45c0-978c-8c4c00f8eb84)) \(d_pN\colon T_p\Sigma \to T_{N(p)}\mathbb{S}^2\) of the Gauss map \(N\): \[ S_p(X) = -d_pN(X). \]

• The eigenvalues \(k_1, k_2\) of shape operator are called the principal curvatures, and the corresponding eigenvectros \(E_1, E_2\) are called the principal directions.

• What is "mean" about the mean curvature? - YouTube
• The mean of the principal curvatures, which is also the average line curvature
• The line curvature of cross section \(\Gamma\) is given by:
  • \[ -\mathbf{v}\cdot S_\Gamma(\mathbf{v}) = -\mathbf{v}\cdot S_\Sigma(\mathbf{v}) \]
  • where \(S_\Sigma\) is the shape operator of the surface, \(S_\Gamma\) is the shape operator of the cross section, and \(\mathbf{v}\) is a tangent vector at a point.
  • \(S_\Gamma(\mathbf{v})\) is in the direction of \(\mathbf{v}\).
• It is also \(\frac{1}{2} \operatorname{tr} S_\Sigma\).

• It is a counterexample to the statement that smooth closed surface with principal curvature, or Gaussian curvature for that matters, less than one everywhere, does not always contain a unit sphere in its interior.
• This is contrary to the two dimensional case in which the unit disc is always contained within a simple smooth regular closed curve with curvature less than or equal to 1 everywhere.

• Given a smooth surface \(\Sigma\), the induced distance between two points \(p, q\in \Sigma\) is given by: \[ d(p,q) := \operatorname{inf}\{\operatorname{length}(\gamma)\mid \gamma\colon [a,b]\to \Sigma, \gamma(a) = p, \gamma(b) = q\}. \]
• And the curve that attains the infimum is called a minimizing curve.

• If \((\Sigma, d)\) is a complete metric space, then for all \(p, q\in \Sigma\), there is a minimizing curve between \(p\) and \(q\).

• For a curve \(\gamma\colon [a,b]\to \Sigma\), the energy of the curve is: \[ E(\gamma) := \frac{1}{2}\int_a^b|\gamma'(t)|^2\,dt. \]
• Energy of a curve depends on parametrization.

• A smooth map \(\theta\colon (-\varepsilon, \varepsilon)\times [a,b]\to \Sigma\) is a variation of \(\gamma\colon [a,b]\to \Sigma\), if \(\forall t\in [a,b],\theta(0,t) = \gamma(t)\).
• \(V(t) := \theta_s(0,t)\) is called the direction of \(\theta\).

• A \(C^2\)-curve \(\gamma\colon I\to \Sigma\) is called a geodesic if: \[ \forall t\in I, \gamma''(t) \perp T_{\gamma(t)}\Sigma. \]
• It minimizes the energy.

• For a naturally parametrized smooth curve \(\gamma\colon I\to \Sigma\) on a oriented surface with Gauss map \(N\), the geodesic curvature is: \[ k_\gamma(t) := \gamma''(t)\cdot(N(\gamma(t))\times \gamma'(t)). \]
• A curve is geodesic precisely when the geodesic curvature is zero everywhere.

• Given a vector field \(V\) along \(\gamma\),
• The vector field \(D_tV := \pi_t(V'(t))\) is called the covariant derivative of \(V\), where \(\pi_t\colon \mathbb{R}^3\to T_{\gamma(t)}\Sigma\) is the orthogonal projection.
  • Acceleration into the manifold.

• For a surface \(\Sigma\subset \mathbb{R}^3\), the following are equivalent:
  • The metric space \((\Sigma, d)\) is complete.
  • For all \(q\in \Sigma\), the map \(\exp_q\colon T_q\Sigma\to \Sigma\) is defined.
  • There is \(p\in \Sigma\) such that the map \(\exp_p\colon T_p\Sigma\to \Sigma\) is defined.

• Motivation: \(\exists B\text{ ball}\in T_p\Sigma\), such that \(s := \exp_p\colon B\to \Sigma\) is a chart.
  • Polar coordinates within \(B\) induce vector fields \(s_r, s_\theta\colon B\setminus\{0\}\to \mathbb{R}^3\).
  • Gauss Lemma: The vector fields \(s_r\) and \(s_\theta\) are orthogonal.
  • \(|s_r| = 1.\)
• A chart \(s\colon U\to \Sigma\) is called a semigeodesic chart if
  • For each \(v\), the map \(u\mapsto s(u,v)\) is a unit speed geodesic.
  • The vectors \(s_u\) and \(s_v\) are orthogonal.

• For a semigeodesic chart \(s\colon U\to \Sigma\), and \(b = |s_v|\): \[ bK + b_{uu} = 0. \]
• \(b\) is the measure of how spread out the geodesics are.
• If the surface is elliptic, then the geodesics tends to converge, on the other hand if the surface is hyperbolic, then the geodesics tends to diverge.

• Latin for "Remarkable Theorem"
• Under the local isomety \(\phi\colon \Sigma_1\to \Sigma_2\), that is,
  • \[ \forall p\in \Sigma_1, \forall X\in T_p\Sigma, |d_p\phi(X)| = |X|, \]
• The 2.4.4 is invariant: \[ \forall p\in \Sigma_1, K_{\Sigma_2}(\phi(p)) = K_{\Sigma_1}(p). \]

• For a compact two-dimensional \(M\) with boundary \(\partial M\), \[ \int_M K\,dA + \int_{\partial M}k_g\,ds = 2\pi\chi(M) \]
• where \(K\) is the 2.4.4 of \(M\), \(k_g\) is the 2.5.6 of \(\partial M\), and \(\chi(M)\) is the of \(M\).
• This implies that the global curvature is invariant.

\( K\,dA \) is the signed area on the unit sphere with the same solid angle. For a each hole the entire surface of the sphere is covered with the direction of the area reversed, so the formula becomes:

where \( T \) is a geodesic triangle, and \( \alpha, \beta, \gamma \) are the interior angles of it.

Notice that the parallel transport is preserved by the differential of the Gauss map, and the holonomy at the triangle on the unit sphere is equal to the area of the triangle.

• The inclusion of the chart into a differentiable chart results in a differentiable atlas.

• Consists of all charts that are differentiably compatible with the given atlas.
• Maximal smooth atlas is called the smooth structure.
• Maximal holomorphic atlas is called the complex structure.

• Tangent vector at \(p\in M\) is an equivalence class of differentiable curves \(\gamma\) with \(\gamma(0) = p\) with the equivalence relation of first-order 1.13
• Tangent vector \(X\) at \(p\) uniquely determines the directional derivative of the differentiable function \(f\) at \(p\): \[ Xf(p) := \frac{d}{dt}f(\gamma(t))\bigg|_{t=0}. \]
  • \(X\) can be thought of as \(\gamma'(0)\).

• Fixing \(f\), the mapping \(X \mapsto Xf(p)\) which is a linear functional on the tangent space, is called the differential of \(f\) at \(p\): \[ d_pf\colon T_pM \to \mathbb{R}. \]

• The image \(d\phi_x(X)\) of the tangent vector \(X\in T_xM\) under \(d\phi_x\) is sometimes called the pushforward of \(X\) by \(\phi\).

• Pushforward Measure, Pushforward, Image Measure

• For a smooth map \(\phi\colon M\to N\) between manifolds \(M\) and \(N\), the pullback of a smooth function \(f\colon N\to \mathbb{R}\) by \(\phi\) is the smooth function \(\phi^*f\colon M\to \mathbb{R}\) defined by \[ \phi^*f(x) = f(\phi(x)). \]

• For a smooth map \(\phi\colon M\to N\) between smooth manifolds \(M\) and \(N\), there is an associated linear map \(d\phi\) from the space of 1-forms on \(N\) to the space of 1-forms on \(M\).

• A 3.4 \(f\colon M\to N\) between 3 \(M\) and \(N\), whose 3.5 is everywhere injective, that is,
  • \[ d_pf\colon T_pM \to T_{f(p)}N \] is an injective function at every point \(p\in M\).
• Equivalently, the derivative of \(f\) has constant rank equal to the dimension of \(M\),
  • \[ \operatorname{rank}d_pf = \dim M. \]
• The function \(f\) itself need not be injective.

• Immersion is presicely a local

• A differentiable map \(f\colon M\to N\) between differentiable manifolds \(M, N\) is a submersion if its 3.5 is everywhere surjective linear.

• The notion of submersion dual to the notion of an 3.8

• A 3.4 \(f\colon M\to N\) is called diffeomorphism if
  • \(f\) is bijection
  • \(f^{-1}\) is differentiable
• \(r\) times continuously differentiable diffeomorphism \(f\) is called \(C^r\)-diffeomorphism.

• If \(f\colon M \to \mathbb{R}^n\) is smooth, its ./Vector and Matrix Calculus.html#orgabf21e2 at \(p\) is defined as: \[ \mathbf{J}_f(p) := \mathbf{J}_{d_pf}. \]

• See Also

• For a \(n\)-form \(\omega\) and \(m\)-form \(\sigma\), \(\wedge: (\omega, \sigma) \mapsto \omega\wedge \sigma\) such that: \[ (\omega\wedge\sigma)(X_1,\dots,X_{n+m}) \vcentcolon= \frac{1}{n!}\frac{1}{m!}\sum_{\pi\in S_{n+m}}\operatorname{sgn}(\pi)(\omega\otimes\sigma)(X_{\pi(1)},\dots,X_{\pi(n+m)}) \] where \(S\) is the ((655459b8-19b7-4009-9a15-f023279bb87a)).

• For 2-forms in three dimension:
  • \[ dx^i\wedge dx^j = \sum_{k, l} \begin{vmatrix} \dfrac{\partial x^i}{\partial \tilde{x}^k} & \dfrac{\partial x^i}{\partial \tilde{x}^l} \\[10pt] \dfrac{\partial x^j}{\partial \tilde{x}^k} & \dfrac{\partial x^j}{\partial \tilde{x}^l} \end{vmatrix} d\tilde{x}^k\wedge d\tilde{x}^l \]
• The determinants is the scaling factors, which becomes the determinant of the Jacobian if the order of the form coincides with the number of dimensions, i.e. the form is a pseudoscalar.

• \[ \bigwedge_i dx^i = \bigwedge_i \frac{\partial x^i}{\partial \tilde{x}^j} d\tilde{x}^j \]
• This can be seen as using the inverse .

• Since differential form is a tensor with special properties, it follows the normal transformation rules of the , which is equivalent to the above two methods.

• \[ d\omega = d(f_Idx^I) = \frac{\partial f_I}{\partial x^j}dx^j\wedge dx^I \] where \(I\) is the multi-index such that \(dx^I = dx^{i_1}\wedge dx^{i_2}\wedge\dots\wedge dx^{i_m}\).

• \[ d(\omega\wedge\mu) = (d\omega)\wedge \mu + (-1)^m\omega\wedge(d\mu) \] where \(\omega\) is an \(m\)-form.
• \[ d^2\omega = 0 \]

• \(\star\), Hodge Star
• It is the complement. Formally: \[ \star dx_I = dx_J: dx_I\wedge dx_J = dx_1\wedge dx_2\wedge \dots \wedge dx_n \] where \(I\) and \(J\) are the multi-indices.
• e.g. In \(\mathbb{R}^3\), \(\star dx = dy\wedge dz\).

• \[ \nabla f \rightsquigarrow df \]

• \[ \nabla \times \mathbf{F} \rightsquigarrow \star d \omega_{F} \]

• \[ \nabla \cdot \mathbf{F} \rightsquigarrow \star d(\star \omega_F) \]

• \(\langle \cdot ,\cdot \rangle: \bigwedge^m(\mathbb{R}^n)\times \bigwedge^m(\mathbb{R}^n)\to \mathbb{R}\).
• Only the inner product for the basis needs to be defined for the entire inner product to be defined, since the lifting.

• Inner product for a lower form can be lifted for a higher form with \[ \langle dx_I, dx_J \rangle = \sum_{\sigma \in S_m}\left(\operatorname{sgn}(\sigma)\prod_{k=1}^m\langle dx_{i_k}, dx_{j_{\sigma(k)}}\rangle\right) \] where \(I\) and \(J\) are multi-indices and \(S_m\) is the set of all permutations of \(m\) elements.

• \(\omega_1\wedge\star\omega_2 = \langle \omega_1, \omega_2\rangle dx_1\wedge dx_2\wedge\dots\wedge dx_n\), where \(\omega_1\) and \(\omega_2\) are of the same order.

• \[ \mathbf{e}_i=\frac{\partial}{\partial x^i} \]
• This works everywhere on a manifold. \(\vec{e}_i\) may change its direction and magnitude.
• \[ \frac{d}{d\lambda}=\frac{\partial}{\partial x^i}\frac{dx^i}{d\lambda} \]

• \[ \frac{\partial}{\partial x^i}\frac{dx^i}{d\lambda}=\frac{\partial}{\partial x^i}J^{-1}\vphantom{J}^i_jJ^j_k\frac{dx^k}{d\lambda}=\frac{\partial}{\partial \tilde{x}^j}\frac{d\tilde{x}^j}{d\lambda} \]
• Basis vectors are covariant and vector components are contravariant.

• \(\delta_{ij}\), \(\delta_i^j\), \(\delta^{ij}\)
• 1 if the indices are the same, 0 if not.

• \(\delta^{\mu_1\dots\mu_p}_{\nu_1\dots\nu_p}\)
  • 1 if \(\nu_i\)s are distinct integers and are an even permutation of \(\mu_i\)s,
  • -1 if \(\nu_i\)s are distinct integers and are an odd permutation of \(\mu_i\)s,
  • 0 otherwise.
• \[ \delta^{\mu_1\dots\mu_p}_{\nu_1\dots\nu_p} = \sum_{\sigma\in S_p}\mathrm{sgn}(\sigma)\delta^{\mu_1}_{\nu_{\sigma(1)}}\cdots \delta^{\mu_p}_{\nu_{\sigma(p)}} = \sum_{\sigma\in S_p}\mathrm{sgn}(\sigma)\delta^{\mu_{\sigma(1)}}_{\nu_1}\cdots \delta^{\mu_{\sigma(p)}}_{\nu_p} \]

• Levi-Civita symbol \(\epsilon_{\mu_1\mu_2\dots\mu_n}\) is \(1\) when \(\mu_i\) are distinct and \((\mu_1\mu_2\dots\mu_n)\) is an even permutaion, and \(-1\) when \(\mu_i\) are distinct and it's an odd permutation, and \(0\) otherwise.

• Using the Einstein notation:
• \(\varepsilon_{ijk} \varepsilon^{pqr}=\delta_i{}^p\delta_j{}^q\delta_k{}^r - \delta_i{}^p\delta_j{}^r\delta_k{}^q + \delta_i{}^r\delta_j{}^p\delta_k{}^q - \delta_i{}^r\delta_j{}^q\delta_k{}^p + \delta_i{}^q\delta_j{}^r\delta_k{}^p - \delta_i{}^q\delta_j{}^p\delta_k{}^r.\)
• \(\varepsilon_{ijk}\varepsilon^{pqk} = \delta_i^p\delta_j^q - \delta_i^q\delta_j^p.\)
• \(\varepsilon_{jmn}\varepsilon^{imn} = 2\delta_j^i.\)
• \(\varepsilon_{ijk}\varepsilon_{ijk}=6.\)

• \[ \epsilon^i=dx^i \]
• \[ df=\frac{\partial f}{\partial x^i}dx^i \]

• \[\frac{\partial f}{\partial x^i}dx^i=\frac{\partial f}{\partial x^i}J^{-1}\vphantom{J}^i_j J^j_k dx^k=\frac{\partial f}{\partial \tilde{x}^j}d\tilde{x}^j\]
• Basis covectors are contravariant and covector components are covariant.

• \[ g_{ij}(\epsilon^i\otimes\epsilon^j)=\frac{\partial}{\partial x^i}\cdot\frac{\partial}{\partial x^j}(\epsilon^i\otimes\epsilon^j) \]

• Bilinear
• Symmetric

• \[ g_{ij}\epsilon^{i}\epsilon^{j}=\frac{\partial}{\partial x^i}\cdot\frac{\partial}{\partial x^j}\epsilon^{i}\epsilon^{j}=\frac{\partial}{\partial x^i}\cdot\frac{\partial}{\partial x^j}J^{-1}\vphantom{J}^i_kJ^{-1}\vphantom{J}^j_lJ^k_{m}J^l_{n}\epsilon^{m}\epsilon^{n}=\frac{\partial}{\partial \tilde{x}^k}\cdot\frac{\partial}{\partial \tilde{x}^l}\tilde{\epsilon}^{k}\tilde{\epsilon}^{l}=\tilde{g}_{kl}\tilde{\epsilon}^{k}\tilde{\epsilon}^{l} \]
• \[ \left(\mathbf{J}^{-1}\right)^\mathrm{T}\mathbf{g}\mathbf{J}^{-1}=\tilde{\mathbf{g}} \]

• \[ \tilde{\Gamma}^k_{ij}=\frac{\partial \tilde{x}^k}{\partial x^n}\left( \frac{\partial x^l}{\partial \tilde{x}^i}\frac{\partial x^m}{\partial \tilde{x}^j} \Gamma^{n}_{lm}+\frac{\partial^2 x^n}{\partial \tilde{x}^i\partial \tilde{x}^j}\right) \]

• A curve with zero tangential acceleration, which means a curve \(C: \lambda \mapsto \mathbf{x}\) satisfying the geodesic equation.

• \[ \frac{d^2}{d\lambda^2} = 0 \]
• Component form \[ \frac{d^2 x^k}{d\lambda^2} + \frac{d x^i}{d\lambda}\frac{d x^j}{d\lambda}\Gamma^k_{ij}=0 \] where \(\Gamma^k_{ij}\) is the 5.8.
• It is about the covariant derivative of the tangent vectors along the curve being equal to zero.

• \[ \nabla_{\partial_i}(\alpha)=\left( \frac{\partial \alpha_k}{\partial x^i} - \alpha_j\Gamma^j_{ik} \right)dx^k \]

• Assert \[ \nabla_{\mathbf{w}}(T\otimes S)=(\nabla_{\mathbf{w}}T)\otimes S+T\otimes(\nabla_{\mathbf{w}}S) \]
• For a (n, m)-Tensor we have \(n\) positive Christoffel symbol terms and \(m\) negative ones.

• Coordinate definition

\[ [X, Y](f) = (X^j\partial_jY^i - Y^j\partial_jX^i)\partial_i \]

• Bianchi Identity: \( R_{dcab} + R_{dbca} + R_{dabc} = 0 \) from torsion-free
• 12-symmetry: \( R_{abcd} = - R_{bacd} \) from metric compatibility.
• 34-symmetry: \( R^a_{bcd} = -R^a_{bdc} \)
• Flip: \( R_{abcd} = R_{cdab} \)

• From the geodesic equation \( \nabla_\mathbf{v}\mathbf{v} = 0 \) and the torsion-free condition, the relation between the geodesic deviation and the Riemann curvature tensor is derived:

\[ \nabla_\mathbf{v}\nabla_\mathbf{v} \mathbf{s} = - R(\mathbf{s},\mathbf{v})\mathbf{v}, \] where \(\mathbf{s}\) is the separation vector between a geodesic and nearby ones, and \(\mathbf{v}\) are the tangent vector along a geodesic.

• The sectional curvature is defined as follows:

\[ K(\mathbf{s}, \mathbf{v}) := \frac{R(\mathbf{s}, \mathbf{v})\mathbf{v} \cdot \mathbf{s}}{(\mathbf{s}\cdot\mathbf{s})(\mathbf{v}\cdot\mathbf{v}) - (\mathbf{s}\cdot\mathbf{v})^2} \]

• The manifold is elliptic, in other words, geodesics converges if this is positive, and otherwise, negative.
• It is independent of the choice of vectors, only on the plane that contains those vectors.

• \[ R=g^{ij}R_{ij} \]
• Positive Ricci scalar means the space is elliptic and negative means hyperbolic.

\[ \omega(\mathbf{u},\mathbf{v},\mathbf{w})=\sqrt{\det(g)}\epsilon_{ijk}u^iv^jw^k \] where \(\epsilon_{ijk}\) is the .

• \[ \nabla_\mathbf{v}V = 0 \implies \nabla_\mathbf{v}\omega = 0 \] using the Levi-Civita connection that preserves the lengths and angles.
• Using separation vectors \( \mathbf{s}_1, \mathbf{s}_2, \mathbf{s}_3 \), and the geodesic deviation \( \ddot{s}_j^{\mu_j} = -R^{\mu_j}_{xyz}s_j^yv^zv^x \), the second derivative of the volume is given by: \[ \frac{d^2V}{d\lambda^2} = -R_{xy}v^xv^z(V) + \left( \dot{s}_j^{\mu_j}\dot{s}_k^{\mu_k}\prod_{i\neq j, k, i=1}^D s_i^{\mu_i} \right) \sqrt{\det g} \epsilon_{\mu_1\dots\mu_D} \]
  • The first term is the acceleration term and the second term is velocity term, therefore the first term tells the curvature.