# Definition:Tangent Space

## Definition

### Real Submanifold

Let $M$ be a real submanifold of $\R^n$ of dimension $d$.

Let $p\in M$.

#### Using Local Submersions

Let $U$ be a open neighborhood of $p$ in $\R^n$ and $\phi : U \to \R^{n-d}$ be a submersion such that:

- $M \cap U = \phi^{-1}(0)$.

The **tangent space of $M$ at $p$** is:

- $T_pM = \ker d\phi(p)$

where $d\phi(p)$ is the differential of $\phi$ at $p$.

#### Using Local Embdeddings

Let $U$ be a open set of $\R^d$ and $\phi : U \to \R^{n}$ be an embedding such that:

- $p\in \phi(U) \subset M$

The **tangent space of $M$ at $p$** is:

- $T_pM = \operatorname{im} (d\phi)(\phi^{-1}(p))$

where $(d\phi)(\phi^{-1}(p))$ is the differential of $\phi$ at $\phi^{-1}(p)$.

#### Using Local Immersions

### Geometric Tangent Space

### Differentiable Manifold

There are various ways to construct the **tangent space** of a differentiable manifold.

Let $M$ be a smooth manifold of dimension $m$.

Let $p \in M$.

#### Abstract Definition

Let $A$ be an atlas of $M$.

Let $B = \{(U,\phi) \in A : p\in U\}$.

The **tangent space of $M$ at $p$** is the real vector space

- $\displaystyle \left(\coprod_{b\in B}\{b\} \times \R^m\right)/\sim$

where $\sim$ is the equivalence relation defined by:

- $(i,\xi) \sim (j,\eta) \iff d(\phi_j \circ \phi_i^{-1})(\phi_i(p)) (\xi) = \eta$

where $d(\phi_j \circ \phi_i^{-1})(\phi_i(p))$ is the differential of $\phi_j \circ \phi_i^{-1}$ at $\phi_i(p)$.

Addition is defined by:

- $[(i,\xi)] + [(j,\eta)] = \left[\left( i, d(\phi_j \circ \phi_i^{-1})(\phi_i(p)) (\xi) + \eta \right)\right]$

and scalar multiplication by:

- $\lambda \cdot [(i,\xi)] = [(i, \lambda\cdot\xi)]$

#### As a space of germs of curves

#### As a space of derivations

The **tangent space at $p$**, denoted by $T_p M$, is the set of all tangent vectors at $m$.

## Equivalence of Definitions

While the above constructions are not the same, there are very closely related. See Equivalence of Definitions of Tangent Space.

## Also see

- Equivalence of Definitions of Tangent Space
- Tangent Space is Vector Space
- Definition:Tangent Vector
- Definition:Tangent Bundle

## Sources

- 2013: Gerd Rudolph and Matthias Schmidt:
*Differential Geometry and Mathematical Physics*: $\S 1.4$: Tangent Space