Tangent Bundle

The Tangent Bundle of a differentiable manifold $M$ is the disjoint union of the tangent spaces at all points in $M$, denoted $T_\ast M$.

Formally,

$$T_\ast M = \bigcup_{p \in M} T_pM$$

The tangent bundle is itself a differentiable manifold, of double the dimension of $M$. Its structure is crucial to the study of vector fields and differential forms.

Topology and Smooth Structure

Tangent bundle is equipped with a natural topology and smooth structure, with $\dim T_\ast M = 2 \dim M$. Each tangent space of an n-manifold is an n-dimensional vector space. If $U$ is open contractible subset of $M$, then there is a diffeomorphism $TU \to U \times \mathbb{R}^n$ which restricts to a linear isomorphism from each tangent space $T_x U$ to $\{x\} \times \mathbb{R}^n$. As a manifold, $T_\ast M$ is not always diffeomorphic to product $M \times \mathbb{R}^n$. When it is, the tangent bundle is said to be a trivial tangent bundle, for instance this is always the case when the manifold is a Lie group. For example, tangent bundle of a unit circle, $T_\ast \mathrm{U}(1)$ is trivial. It is not true however that all spaces with a trivial tangent bundles are Lie groups. Manifolds with trivial tangent bundles are called parallelizable manifolds.

Lifts

There are various ways to lift objects on $M$ into $T_\ast M$. For example, let $\gamma$ be a curve in $M$, then $\gamma'$ (the tangent of $\gamma$) is a curve in $T_\ast M$.

The vertical lift of a function $f : M \to \mathbb{R}$ is the function $f^\wedge T_\ast M \to \mathbb{R}$ defined by $f^\wedge = f \circ \pi$, where $\pi : T_\ast M \to M$ is the canonical projection.

In contrast to tangent bundles, without further assumptions, there is no similar lift into the cotangent bundle in general. However, Riemannian manifolds obtain a similar lift into the cotangent bundle.

In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. The canonical isomorphism induced by metric tensor on a pseudo-Riemannian or a symplectic manifold is referred to as a musical isomorphism.

Higher-order tangent bundles

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Vector fields

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Canonical vector field

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Unit tangent bundle

In Riemannian geometry, the unit tangent bundle of a Riemannian manifold $(M, g)$, denoted $\overline{T_\ast M}$ is the unit sphere bundle for the tangent bundle $T_\ast M$, it is a fiber bundle over $M$ whose fiber at each point is the unit sphere in the tangent bundle:

$$\overline{T_\ast M} = \coprod_{x \in M} \{ \vec{v} \in T_x(M)\; | \; g_x(\vec{v},\vec{v}) = 1\}$$

Elements of $\overline{T_\ast M}$ are pairs $(x,\vec{v})$, where $x$ is a point on a manifold $M$ and $\vec{v}$ is a unit vector in the tangent space of that point $T_x M$. The unit tangent bundle is equipped with a natural projection onto the base space $\pi : \overline{T_\ast M} \to M$, $\pi : (x,\vec{v}) \mapsto x$. The fiber $\pi^{-1}(x)$ over each point $x \in M$ is a (n-1)-dimensional hypersphere $S^{n-1}$, (where n is dimension of the base space), and is therefore a sphere bundle over $M$ with a fiber $S^{n-1}$.

Geometric Structure

Unit tangent bundle carries a variety of differential geometric structures. Metric on $M$ induces a contact structure on $\overline{T_\ast M}$ given in terms of tautological one-form, (a special 1-form $\theta$ of a cotangent bundle $T^\ast M$ that "cancels" $\beta^\ast \theta = \beta$ for every 1-form $\beta$ on $M$), defined at a $\vec{u} \in \overline{T_\ast M}$ as $\theta_u(\vec{v}) = g(\vec{u}, \pi_\ast \vec{v})$. Geometrically, this contact structure is the distribution of (2n-2)-planes which at the unit vector $\vec{u}$ is the orthogonal complement of $u$ in $T_\ast M$. The volume form $\theta \wedge d\theta^{n-1}$ defines a kinematic measure, or Liouville measure on $M$ that is invariant under the geodesic flow of $M$. The kinematic measure $\mu$ is defined on compactly supported continuous functions $f$ on $\overline{T_\ast M}$ by:

$$\int_{\overline{T_\ast M}} f\; d\mu = \int_M dV(x) \int_{\overline{T_x M}} f |_{\overline{T_\ast M}}\; d\mu_x$$

where $dV$ is the volume element on $M$, and $\mu_x$ is the standard rotationally-invariant Borel measure on the Euclidean sphere $\overline{T_x M}$.

The Levi-Civita connection of $M$ then gives a rise to a splitting of the tangent bundle $T_\ast \overline{T_\ast M} = H \oplus V$ into a vertical space $V = \mathrm{ker} \pi_\ast$ and horizontal space $H$ on which $\pi_\ast$ is a linear isomorphism at each point of $\overline{T_\ast M}$. Splitting induces a metric on $\overline{T_\ast M}$ given as an orthogonal direct sum. Metric on $H$ is given by the pullback $g_H(\vec{v},\vec{w}) = g(\vec{v},\vec{w})$ for $\vec{v}, \vec{w} \in H$ and metric on $V$ is induced from the embedding of the unit fiber $\overline{T_x M}$ into the Euclidian tangent space $T_x M$. A manifold equipped with this metric is called a Sasakian manifold.