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In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics, the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.
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In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics, the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.
Contents
Informal description
In differential geometry, one can attach to every point
x
{\displaystyle x}
of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through
x
{\displaystyle x}
. The elements of the tangent space at
x
{\displaystyle x}
are called the tangent vectors at
x
{\displaystyle x}
. This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself.
For example, if the given manifold is a
2
{\displaystyle 2}
-sphere, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport. Many authors in differential geometry and general relativity use it. More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.
Formal definitions
The informal description above relies on a manifold's ability to be embedded into an ambient vector space
R
m
{\displaystyle \mathbb {R} ^{m}}
so that the tangent vectors can "stick out" of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself.
There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.
Definition via tangent curves
In the embedded-manifold picture, a tangent vector at a point
x
{\displaystyle x}
is thought of as the velocity of a curve passing through the point
x
{\displaystyle x}
. We can therefore define a tangent vector as an equivalence class of curves passing through
x
{\displaystyle x}
while being tangent to each other at
x
{\displaystyle x}
.
Suppose that
M
{\displaystyle M}
is a
C
k
{\displaystyle C^{k}}
differentiable manifold (with smoothness
k
≥
1
{\displaystyle k\geq 1}
) and that
x
∈
Definition via derivations
Suppose now that
M
{\displaystyle M}
is a
C
∞
{\displaystyle C^{\infty }}
manifold. A real-valued function
f
:
M
→
R
{\displaystyle f:M\to \mathbb {R} }
is said to belong to
C
∞
(
M
)
{\displaystyle {C^{\infty }}(M)}
if and only if for every coordinate chart
φ
:
U
→
R
n
{\displaystyle \varphi :U\to \mathbb {R} ^{n}}
, the map
f
∘
Equivalence of the definitions
For
x
∈
M
{\displaystyle x\in M}
and a differentiable curve
γ
:
(
−
1
,
1
)
→
M
{\displaystyle \gamma :(-1,1)\to M}
such that
γ
(
0
)
=
x
,
{\displaystyle \gamma (0)=x,}
define
D
γ
(
f
)
:=
(
f
∘
Definition via cotangent spaces
Again, we start with a
C
∞
{\displaystyle C^{\infty }}
manifold
M
{\displaystyle M}
and a point
x
∈
M
{\displaystyle x\in M}
. Consider the ideal
I
{\displaystyle I}
of
C
∞
(
M
)
{\displaystyle C^{\infty }(M)}
that consists of all smooth functions
f
{\displaystyle f}
vanishing at
x
{\displaystyle x}
, i.e.,
f
(
x
Properties
If
M
{\displaystyle M}
is an open subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
, then
M
{\displaystyle M}
is a
C
∞
{\displaystyle C^{\infty }}
manifold in a natural manner (take coordinate charts to be identity maps on open subsets of
R
n
{\displaystyle \mathbb {R} ^{n}}
), and the tangent spaces are all naturally identified with
R
n
{\displaystyle \mathbb {R} ^{n}}
.
Tangent vectors as directional derivatives
Another way to think about tangent vectors is as directional derivatives. Given a vector
v
{\displaystyle v}
in
R
n
{\displaystyle \mathbb {R} ^{n}}
, one defines the corresponding directional derivative at a point
If the tangent space is defined via differentiable curves, then this map is defined by
d
φ
x
(
γ
′
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
In algebraic geometry, in contrast, there is an intrinsic definition of the tangent space at a point of an algebraic variety
V
{\displaystyle V}
that gives a vector space with dimension at least that of
V
{\displaystyle V}
itself. The points
p
{\displaystyle p}
at which the dimension of the tangent space is exactly that of
V
{\displaystyle V}
are called non-singular points; the others are called singular points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of
V
{\displaystyle V}
are those where the "test to be a manifold" fails. See Zariski tangent space.
Once the tangent spaces of a manifold have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: A solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.
All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called the tangent bundle of the manifold.
Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties. However, instead of examining derivations
D
{\displaystyle D}
from the full algebra of functions, one must instead work at the level of germs of functions. The reason for this is that the structure sheaf may not be fine for such structures. For example, let
is a vector space isomorphism between the space of the equivalence classes
γ
′
(
0
)
{\displaystyle \gamma '(0)}
and the space of derivations at the point
x
.
{\displaystyle x.}
)
=
0
{\displaystyle f(x)=0}
. Then
I
{\displaystyle I}
and
I
2
{\displaystyle I^{2}}
are both real vector spaces, and the quotient space
I
/
I
2
{\displaystyle I/I^{2}}
can be shown to be isomorphic to the cotangent space
T
x
∗
M
{\displaystyle T_{x}^{*}M}
through the use of Taylor's theorem. The tangent space
T
x
M
{\displaystyle T_{x}M}
may then be defined as the dual space of
I
/
I
2
{\displaystyle I/I^{2}}
.
While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the varieties considered in algebraic geometry.
is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.
As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if
v
{\displaystyle v}
is a tangent vector to
M
{\displaystyle M}
at a point
x
{\displaystyle x}
(thought of as a derivation), then define the directional derivative
