Tangent space

\scriptstyle T_{x}M

{\ displaystyle \ scriptstyle T_ {x} M}

\ scriptstyle T_ {x} M

and tangent vector

\scriptstyle v\in T_{x}M

{\ displaystyle \ scriptstyle v \ in T_ {x} M}

\ scriptstyle v \ in T_ {x} M

along the curve

\scriptstyle \gamma (t)

{\ displaystyle \ scriptstyle \ gamma (t)}

\ scriptstyle \ gamma (t)

passing through a point

\scriptstyle x\in M

{\ displaystyle \ scriptstyle x \ in M}

\ scriptstyle x \ in M

Tangent space to a smooth manifold $M$ ${\ displaystyle M}$ $M$ at the point $x$ ${\ displaystyle x}$ $x$ - a set of tangent vectors with the natural structure of a vector space introduced on it. Tangent space to $M$ ${\ displaystyle M}$ $M$ at the point $x$ ${\ displaystyle x}$ $x$ usually indicated $T_{x}M$ ${\ displaystyle T_ {x} M}$ $T_ {x} M$ or - when it’s obvious what kind of variety is involved - just $T_{x}$ ${\ displaystyle T_ {x}}$ $T_ {x}$ .

The set of tangent spaces at all points of the manifold (together with the manifold itself) forms a vector bundle . Accordingly, each tangent space is a layer of the tangent bundle.

Tangent space at a point $p$ ${\ displaystyle p}$ $p$ to a submanifold is defined similarly.

In the simplest case, when a smooth manifold is smoothly embedded in a vector space (which is always possible, according to the Whitney embedding theorem ), each tangent space can naturally be identified with some affine subspace of the ambient vector space.

Content

Definitions

There are two standard definitions of a tangent space: through the equivalence class of smooth curves and through differentiation at a point. The first is intuitively simpler, but a number of technical difficulties arise along the way. The second is the simplest, although the level of abstraction in it is higher. The second definition is also easier to put into practice.

As an equivalence class of smooth curves

Let be $M$ ${\ displaystyle M}$ - smooth variety and $p\in M$ ${\ displaystyle p \ in M}$ . Consider the class $\Gamma _{p}$ ${\ displaystyle \ Gamma _ {p}}$ smooth curves $\gamma \colon \mathbb {I} \to M$ ${\ displaystyle \ gamma \ colon \ mathbb {I} \ to M}$ such that $\gamma (0)=p$ ${\ displaystyle \ gamma (0) = p}$ . We introduce on $\Gamma _{p}$ ${\ displaystyle \ Gamma _ {p}}$ equivalence relation: $\gamma _{1}\sim \gamma _{2}$ ${\ displaystyle \ gamma _ {1} \ sim \ gamma _ {2}}$ if a

|\gamma _{1}(t)-\gamma _{2}(t)|=o(t),t\to 0

{\ displaystyle | \ gamma _ {1} (t) - \ gamma _ {2} (t) | = o (t), t \ to 0}

in some (and therefore in any) map containing $p$ ${\ displaystyle p}$ .

Tangent space elements $T_{p}$ ${\ displaystyle T_ {p}}$ defined as $\sim$ ${\ displaystyle \ sim}$ equivalence classes $\Gamma _{p}$ ${\ displaystyle \ Gamma _ {p}}$ ; i.e

T_{p}=\Gamma _{p}/\sim

{\ displaystyle T_ {p} = \ Gamma _ {p} / \ sim}

.

In a map such that $p$ ${\ displaystyle p}$ corresponds to the origin, curves from $\Gamma _{p}$ ${\ displaystyle \ Gamma _ {p}}$ can be added and multiplied by a number as follows

(\gamma _{1}+\gamma _{2})(t)=\gamma _{1}(t)+\gamma _{2}(t)

{\ displaystyle (\ gamma _ {1} + \ gamma _ {2}) (t) = \ gamma _ {1} (t) + \ gamma _ {2} (t)}

(k\cdot \gamma )(t)=\gamma (k\cdot t)

{\ displaystyle (k \ cdot \ gamma) (t) = \ gamma (k \ cdot t)}

In this case, the result remains in $\Gamma _{p}$ ${\ displaystyle \ Gamma _ {p}}$ .

These operations continue to equivalence classes. $T_{p}=\Gamma _{p}/\sim$ ${\ displaystyle T_ {p} = \ Gamma _ {p} / \ sim}$ . Moreover, induced on $T_{p}$ ${\ displaystyle T_ {p}}$ operations no longer depend on the choice of card. So on $T_{p}$ ${\ displaystyle T_ {p}}$ the structure of the vector space is determined.

Through differentiation at a point

Let be $M$ ${\ displaystyle M}$ - $C^{\infty }$ ${\ displaystyle C ^ {\ infty}}$ -Smooth variety. Then the tangent space to the manifold $M$ ${\ displaystyle M}$ at the point $p\in M$ ${\ displaystyle p \ in M}$ called the space of differentiations at this point, that is, the space of operators $X,$ ${\ displaystyle X,}$ matching each smooth function $f:M\to \mathbb {R}$ ${\ displaystyle f: M \ to \ mathbb {R}}$ number $X f,$ ${\ displaystyle Xf,}$ and satisfying the following two conditions:

$\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ -linearity: $X(\lambda f+\mu h)=\lambda Xf+\mu Xh,\;\lambda ,\mu \in \mathbb {R} ,f,h\in C^{\infty }(M)$ ${\ displaystyle X (\ lambda f + \ mu h) = \ lambda Xf + \ mu Xh, \; \ lambda, \ mu \ in \ mathbb {R}, f, h \ in C ^ {\ infty} (M)}$
Leibniz rule : $X(fh)=(Xf)\cdot h(p)+f(p)\cdot (Xh),\;f,h\in C^{\infty }(M).$ ${\ displaystyle X (fh) = (Xf) \ cdot h (p) + f (p) \ cdot (Xh), \; f, h \ in C ^ {\ infty} (M).}$

On the set of all derivations at the point $p$ ${\ displaystyle p}$ a natural structure of linear space arises:

$(X+Y)f=Xf+Yf;$ ${\ displaystyle (X + Y) f = Xf + Yf;}$
$(k\cdot X)f=k\cdot (Xf).$ ${\ displaystyle (k \ cdot X) f = k \ cdot (Xf).}$

Remarks

When $C^{k}$ ${\ displaystyle C ^ {k}}$ -smooth manifolds, in the definition through differentiation one more property should be added
$Xf=0$ ${\ displaystyle Xf = 0}$ if a $f(q)=o(|p-q|)$ ${\ displaystyle f (q) = o (| pq |)}$

in some (and therefore in any) map containing

p

{\ displaystyle p}

.

Otherwise, this definition will give infinite-dimensional space, including tangent space. This space is sometimes called an algebraic tangent space . See below.

Let be $\gamma \in \Gamma _{p}$ ${\ displaystyle \ gamma \ in \ Gamma _ {p}}$ . Then the Leibniz rule and the linearity condition of the operator are satisfied for $Xf=(f\circ \gamma )'(0)$ ${\ displaystyle Xf = (f \ circ \ gamma) '(0)}$ . This allows us to identify tangent spaces obtained in the first and second definitions.

Properties

Tangent space $n$ ${\ displaystyle n}$ -dimensional smooth manifold is $n$ ${\ displaystyle n}$ -dimensional vector space
For the selected local map $x_{1},\dots ,x_{n}$ ${\ displaystyle x_ {1}, \ dots, x_ {n}}$ , operators $X_{i}$ ${\ displaystyle X_ {i}}$ differentiation by $x_{i}$ ${\ displaystyle x_ {i}}$ :
$X_{i}f={\frac {\partial f}{\partial x_{i}}}(p)$ ${\ displaystyle X_ {i} f = {\ frac {\ partial f} {\ partial x_ {i}}} (p)}$

constitute the basis

T_{p}

{\ displaystyle T_ {p}}

called the holonomic basis .

Related Definitions

A contact element to a manifold at some point is any hyperplane of the tangent space at this point.

Variations and generalizations

Algebraic tangent space

An algebraic tangent space arises when we in the definition of a tangent vector refuse the additional requirement stated in the remark above (which, however, is only relevant for $C^{k}$ ${\ displaystyle C ^ {k}}$ -differentiable manifolds, $k<\infty$ ${\ displaystyle k <\ infty}$ ) Its definition is generalized to any locally ringed space (in particular, to any algebraic variety ).

Let be $M$ ${\ displaystyle M}$ - $C^{k}$ ${\ displaystyle C ^ {k}}$ -differentiable manifold, $C^{k}(M)$ ${\ displaystyle C ^ {k} (M)}$ Is the ring of differentiable functions from $M$ ${\ displaystyle M}$ at $\mathbb {R}$ {\ displaystyle \ mathbb {R}} . Consider the ring $C_{x}^{k}$ ${\ displaystyle C_ {x} ^ {k}}$ function germs at a point $x\in M$ ${\ displaystyle x \ in M}$ and canonical projection $[-]_{x}:C^{k}(M)\to C_{x}^{k}$ ${\ displaystyle [-] _ {x}: C ^ {k} (M) \ to C_ {x} ^ {k}}$ . Denote by ${\mathfrak {m}}_{x}$ ${\ displaystyle {\ mathfrak {m}} _ {x}}$ core of ring homomorphism $[f]_{x}\mapsto f(x)$ ${\ displaystyle [f] _ {x} \ mapsto f (x)}$ . We introduce on $C_{x}^{k}$ ${\ displaystyle C_ {x} ^ {k}}$ structure of a real algebra using injective homomorphism $i:\mathbb {R} \to C_{x}^{k}$ ${\ displaystyle i: \ mathbb {R} \ to C_ {x} ^ {k}}$ , $i(a)=[\mathrm {const} _{a}]_{x}$ ${\ displaystyle i (a) = [\ mathrm {const} _ {a}] _ {x}}$ and we will further identify $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ and $i(\mathbb {R} )$ ${\ displaystyle i (\ mathbb {R})}$ . Equality holds $C_{x}^{k}=\mathbb {R} \oplus {\mathfrak {m}}_{x}$ ${\ displaystyle C_ {x} ^ {k} = \ mathbb {R} \ oplus {\ mathfrak {m}} _ {x}}$ ^[1] . Denote by $C_{x,0}^{k}$ ${\ displaystyle C_ {x, 0} ^ {k}}$ subalgebra $C_{x}^{k}$ ${\ displaystyle C_ {x} ^ {k}}$ consisting of all germs whose representatives have zero differentials at the point $x$ ${\ displaystyle x}$ in each card ; denote $C_{x,d}^{k}=\mathbb {R} \oplus {\mathfrak {m}}_{x}^{2}$ ${\ displaystyle C_ {x, d} ^ {k} = \ mathbb {R} \ oplus {\ mathfrak {m}} _ {x} ^ {2}}$ . notice, that $C_{x,d}^{k}\subset C_{x,0}^{k}$ ${\ displaystyle C_ {x, d} ^ {k} \ subset C_ {x, 0} ^ {k}}$ .

Consider two vector spaces:

$T_{x}M:=(C_{x}^{k}/C_{x,0}^{k})^{*}$ ${\ displaystyle T_ {x} M: = (C_ {x} ^ {k} / C_ {x, 0} ^ {k}) ^ {*}}$ - this space has dimension $\operatorname {dim} M$ ${\ displaystyle \ operatorname {dim} M}$ and coincides with the tangent space to $M$ ${\ displaystyle M}$ at the point $x$ ${\ displaystyle x}$ ,
$(C_{x}^{k}/C_{x,d}^{k})^{*}\cong ({\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2})^{*}$ ${\ displaystyle (C_ {x} ^ {k} / C_ {x, d} ^ {k}) ^ {*} \ cong ({\ mathfrak {m}} _ {x} / {\ mathfrak {m}} _ {x} ^ {2}) ^ {*}}$ - this space is isomorphic to the space of differentiations $C_{x}^{k}=\mathbb {R} \oplus {\mathfrak {m}}_{x}$ ${\ displaystyle C_ {x} ^ {k} = \ mathbb {R} \ oplus {\ mathfrak {m}} _ {x}}$ with values in $\mathbb {R} \subset C_{x}^{k}$ ${\ displaystyle \ mathbb {R} \ subset C_ {x} ^ {k}}$ , it is called an algebraic tangent space ^[2] $M$ ${\ displaystyle M}$ at the point $x$ ${\ displaystyle x}$ .

If a $k<\infty$ ${\ displaystyle k <\ infty}$ then ${\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2}$ ${\ displaystyle {\ mathfrak {m}} _ {x} / {\ mathfrak {m}} _ {x} ^ {2}}$ has a continuum dimension, and $({\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2})^{*}$ ${\ displaystyle ({\ mathfrak {m}} _ {x} / {\ mathfrak {m}} _ {x} ^ {2}) ^ {*}}$ contains $T_{x}M$ ${\ displaystyle T_ {x} M}$ as a nontrivial subspace; when $k=\infty$ ${\ displaystyle k = \ infty}$ or $k=\omega$ ${\ displaystyle k = \ omega}$ these spaces coincide (and $C_{x,0}^{k}=C_{x,d}^{k}$ ${\ displaystyle C_ {x, 0} ^ {k} = C_ {x, d} ^ {k}}$ ) ^[3] . In both cases $T_{x}M$ ${\ displaystyle T_ {x} M}$ can be identified with the (sub) space of differentiations $C_{x}^{k}$ ${\ displaystyle C_ {x} ^ {k}}$ with values in $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ for vector $X\in T_{x}M$ ${\ displaystyle X \ in T_ {x} M}$ formula $X(f)=X([f]_{x})$ ${\ displaystyle X (f) = X ([f] _ {x})}$ defines an injective homomorphism $T_{x}M$ ${\ displaystyle T_ {x} M}$ into the space of differentiations $C^{k}(M)$ ${\ displaystyle C ^ {k} (M)}$ with values in $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ (structure of real algebra on $C^{k}(M)$ ${\ displaystyle C ^ {k} (M)}$ is set similarly $C_{x}^{k}$ ${\ displaystyle C_ {x} ^ {k}}$ ) Moreover, in the case $k=\infty$ ${\ displaystyle k = \ infty}$ it turns out exactly the definition given above.

Notes

↑ J.-P. Serre , Lie Algebras and Lie Groups, M.: Mir, 1969.
↑ Laird E. Taylor , The Tangent Space to a $C^{k}$ ${\ displaystyle C ^ {k}}$ Manifold, Bulletin of AMS, vol. 79, no. 4, July 1973.
↑ JE Marsden, T Ratiu, R Abraham , Manifolds, Tensor Analysis, and Applications, Addison-Wesley Pub. Co., 1983.

[1] J.-P. Serre , Lie Algebras and Lie Groups, M.: Mir, 1969.

[2] Laird E. Taylor , The Tangent Space to a $C^{k}$ ${\ displaystyle C ^ {k}}$ Manifold, Bulletin of AMS, vol. 79, no. 4, July 1973.

[3] JE Marsden, T Ratiu, R Abraham , Manifolds, Tensor Analysis, and Applications, Addison-Wesley Pub. Co., 1983.