Tags: #geomtop/manifolds #projects/notes/reading #todo/learning/definitions #geomtop/Riemannian-geometry

Refs: manifold

Definitions for Manifolds

Definition (Tangent Bundle): \(TM = {\textstyle\coprod}_{p\in M} T_pM\), which fits into the vector bundle \({\mathbf{R}}^n \to TM \to M\) so \(T_p M \cong {\mathbf{R}}^n\).

\(T_p M = \mathop{\mathrm{span}}_{\mathbf{R}}\left\{{\partial x_i}\right\}\)

Definition (Cotangent Bundle): Since \(T_p M\) is a vector space, we can consider its dual \(T_p {}^{ \vee }M\), and similarly the cotangent bundle \({\mathbf{R}}^n \to T {}^{ \vee }M \to M\).

\(T_p {}^{ \vee }M = \mathop{\mathrm{span}}_{\mathbf{R}}\left\{{dx_i}\right\}\).

Definition (Derivative of a Map): Recall that for any smooth function \(H: M\to N\), the derivative of \(H\) at \(p\in M\) is defined by \(dH_p: T_pM \to T_p N\) which we define using the derivation definition of tangent vectors: given a derivation \(v\in T_p M: C^\infty(M) \to {\mathbf{R}}\), we send it to the derivation \(w_v \in T_{q}M = C^\infty(M) \to {\mathbf{R}}\) where \(w_v\) acts on on \(f\in C^\infty(M)\) by precomposition, i.e. \(w_v \curvearrowright f = v(f \circ H)\).

Definition: Fields and Forms A section of \(TM\) is a vector field, and a section of \(T {}^{ \vee }M\) is a 1-form.

More generally, differential \(k{\hbox{-}}\)forms are in \(\Omega^k(M) \coloneqq\Gamma(\Lambda^k T {}^{ \vee }M)\), i.e. sections of exterior powers of the cotangent bundle.

Definition (Closed and Exact Unsorted/differential forms): Let \(d_p: \Omega^p(M) \to \Omega^{p+1}(M)\) be the exterior derivative. Then a form \(\omega\) is closed (or is a cocycle) iff \(\omega \in \ker d_p\), and exact (or a coboundary) iff \(\omega \in \operatorname{im}d_{p-1}\).

Note that closed forms are exact, since \(d^2 = 0\), i.e. \(\omega\) closed implies \(\omega = d\lambda\) implies \(d\omega = d^2 \lambda = 0\) implies \(\omega\) is exact.

If \(\alpha, \beta \in \Omega^p(M)\) with \(\alpha-\beta\) exact, they are said to be cohomologous.

Definition (Vector Field): A vector field \(X\) on \(M\) is a section of the tangent bundle \(TM \xrightarrow{\pi} M\). Recall that these form an algebra \(\mathfrak{X}(M)\) under the Lie bracket.

Note that vector fields can be time-dependent as a section of \(T(M\times I) \to M\times I\).

Definition (Regular Value): Let \(H: M \to {\mathbf{R}}\) be a smooth function, then \(c\in {\mathbf{R}}\) is a regular value iff for every \(p\in H^{-1}(c)\), the induced map \(H^*: T_pM \to T_P {\mathbf{R}}\) is surjective.

Definition (Closed Orbit): An closed orbit of a vector field \(X\) on \(M\) is an element in the loop space \(\gamma \in \Omega M\) (equivalently a map \(\gamma: S^1 \to M\)) satisfying the ODE \({\frac{\partial \gamma}{\partial t}\,}(t) = X(\gamma(t))\).

In words: the ODE says that the tangent vector at every point along the loop \(\gamma\) should precisely be the tangent vector that the vector field \(X\) prescribes at that point.

Note: Every fixed point of \(X\) is trivially a closed orbit.

Definition (Flow): A flow is a group homomorphism \({\mathbf{R}}\to \mathrm{Diff}(M)\) given by \(t\mapsto \phi_t\). Its integral curves are given by \(\gamma_p: {\mathbf{R}}\to M\) where \(t\mapsto \phi_t(p)\).

Remark: Note that \(X(p) \in T_pM\) is a tangent vector at each point, so we can ask that \({\frac{\partial \phi_t}{\partial t}\,} (p) = X(\phi_t(p))\), i.e. that the tangent vectors to a flow are given by a vector field. This works locally, and can be extended globally if \(X\) is compactly supported.

Definition (Interior Product): Let \(M\) be a manifold and \(X\) a vector field. The interior product is a map

\begin{align*} \iota_X: \Omega^{p+1}(M) &\to \Omega^p(M) && & \\ \omega & \mapsto \iota_X \omega: \Lambda^p TM \to {\mathbf{R}}\\ & (X_1, \cdots, X_p) \to \omega (\mathbf{X}, X_1, \cdots, X_p) \end{align*}

Note that this contracts a vector field with a differential form, coming from a natural pairing on \((i, j)\) tensors \(V^{\otimes i}\otimes(V {}^{ \vee })^{\otimes j}\).

Definition (Lie Derivative):

General definition: For an arbitrary tensor field \(T\) (a section of some tensor bundle \(V \to TM^{\otimes n} \to M\), example: Riemann curvature tensor, or any differential form) and a vector field \(X\) (a section of the tangent bundle \(W \to TM \to M\)), we can define a “derivative” of \(T\) along \(X\). Namely, \begin{align*} (\mathcal{L}_X T)_p = \left[{\frac{\partial }{\partial t}\,} \qty{(\phi_{-t})_* T_{\phi_t(p)}} \right]_{t=0} \end{align*} where

• \(\phi_t\) is the 1-parameter group of diffeomorphisms induced by the flow induced by \(X\),
• \(({-})_*\) is the pushforward

This measures how a tensor field changes as we flow it along a vector field.

Specialized definition: If \(\omega \in \Omega^{p+1}(M)\) is a differential form, we define \begin{align*} \mathcal{L}_x\omega = [d, \iota_x] \omega = d(\iota_x \omega) - \iota_x(d\omega) \end{align*} where \(d\) is the exterior product.

This is a consequence of “Cartan’s Magic Formula”, not the actual definition!

Symplectic

Definition (Symplectic Vector Field): A vector field \(X\) is symplectic iff \(\mathcal{L}_X(\omega) = 0\).

Remark: Then the flow \(\phi_X\) preserves the symplectic structure.

Definition (Hamiltonian Vector Field): If \(X\) is a vector field and \(\iota_X \omega\) is both closed and exact, then \(X\) is a Hamiltonian vector field.

Definition (Exact Symplectic Manifold) Exact symplectic manifold \((M, \lambda)\), \(\lambda \in \Omega^1(M)\), with \(\omega = d\lambda\) for \(\omega \in \Omega^2(M)\).

Definition (Liouville vector field} Liouville vector field \(X\) is the solution of \({\mathcal{L}}_X \omega = \omega\), which yields \(i_x d\lambda = \lambda\) where \(i_x \omega(y) \coloneqq\omega(x, y)\).

Contact

Definition (Overtwisted Contact Structure): \((M, \xi)\) is overtwisted iff there exists an embedded disc \(D^n \xrightarrow{i} M\) such that \(T({{\partial}}D^n)_p \subset \xi_p\) pointwise for all \(p \in {{\partial}}D^n\) and \(TD^n_p\) is transverse to \(\xi\) for every \(p\in (D^n)^\circ\).

Handles

Definition (Normal Bundle): Let \(i: S \hookrightarrow M\) be an embedding, and let \(N_M(S)\) denote the normal bundle of \(S\) in \(M\), which fits into an exact sequence \begin{align*} 0 \to TS \to i^* TM \to N_M(S) \to 0 ,\end{align*}

where \(i*TM\) is the pullback:

so we can identify \(N_M(S) \cong {\left.{{TM}} \right|_{{i(S)}} } /TS\).

Remark: We can “symplectify” this definition by requiring that the pullback of \(\omega\) is constant rank.

Definition (Tubular Neighborhood): For \(S\hookrightarrow M\) an embedded submanifold, a tubular neighborhood of \(S\) is an embedding of the total space of a vector bundle \(E \to S\) along with a smooth map \(J: E \to M\) making the following diagram commute:

where \(0_E\) is the zero section.