---
date: 2022-01-15 21:49
modification date: Saturday 15th January 2022 21:49:49
title: tangent bundle
aliases: [tangent bundle, "tangent space", "cotangent space", "cotangent bundle", exponential, logarithm]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [vector bundle](Unsorted/vector%20bundles.md)
	- [de Rham cohomology](Unsorted/algebraic%20de%20Rham%20cohomology.md)
	- [Riemannian metric](Unsorted/metric.md)
	- [Hermitian K theory](Hermitian%20K%20theory)

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# Tangent space

For schemes: $\T_X = \Sch\slice k(k[\eps]\to X)$, and the tangent space of a functor $F$ is $\T_F \da F(k[\eps])$.

# tangent bundle

Tangent vectors:

![](attachments/Pasted%20image%2020220424180706.png)

Exterior powers:
![](attachments/Pasted%20image%2020220424180803.png)

- $\T_X = \spanof_k\ts{\del x_1, \cdots, \del x_n}$ and $\T_X\dual = \spanof_k \ts{dx_1, \cdots, dx_n}$.
	- This uses the identification $\T_{X} \cong \Der_k(C^\infty(X; \RR), \RR)$ as a space of [derivations](derivations), i.e. $D(xy) = D(x) y + x D(y)$.

![](attachments/Pasted%20image%2020220221004134.png)
![](attachments/Pasted%20image%2020220403173357.png)

![](attachments/Pasted%20image%2020220221004200.png)
![](attachments/Pasted%20image%2020220221004305.png)


# Cotangent bundle

![](attachments/Pasted%20image%2020220424180722.png)


There is a pointwise pairing $\globsec{\T M} \times \globsec{\T\dual M} \to C^\infty(M; \RR)$ where $v,\alpha \mapsto \alpha(v)$.


# Orientability

![](attachments/Pasted%20image%2020220424181000.png)

# Exponentials

![](attachments/2023-03-10exponent.png)