What is tangent space: Definition and 41 Discussions

In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two 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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  1. cianfa72

    I Identification tangent bundle over affine space with product bundle

    Hi, as in this thread Newton Galilean spacetime as fiber bundle I'd like to clarify some point about tangent bundle for an Affine space. As said there, I believe the tangent space ##T_pE## at every point ##p## on the affine space manifold ##E## is canonically/naturally identified with the...
  2. cianfa72

    I Definition of tangent vector on smooth manifold

    I would ask for a clarification about the following definition of tangent vector from J. Lee - Introduction to Smooth Manifold. It applies to Euclidean space ##R^n## with associated tangent space ##R_a^n## at each point ##a \in R^n##. $$D_v\left. \right|_a (f)=D_vf(a)=\left. \frac {df(a + tv)}...
  3. A

    A About computing the tangent space at 1 of certain lie groups

    Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...
  4. C

    I Carroll GR: Tangent Space & Partial Derivatives

    He draws an n-manifold M, a coordinate chart φ : M → Rn, a curve γ : R → M, and a function f : M → R, and wants to specify ##\frac d {d\lambda}## in terms of ##\partial_\mu##. ##\lambda## is the parameter along ##\gamma##, and ##x^\mu## the co-ordinates in ##\text{R}^n##. His first equality is...
  5. U

    I Fixing orientation by fixing a frame in a tangent space

    I would like to show that fixing the orientation of k-manifold smooth connected ##S## in ##\mathbb {R} ^ n ## is equivalent to fixing a frame for one of its tangent spaces. What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with...
  6. S

    B Components of Tangent Space Vector on Parametrized Curve

    I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'. To give some background, I'm aware that basis vectors in tangent space are given by...
  7. S

    I Tangent space basis vectors under a coordinate change

    I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: $$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...
  8. mnb96

    I Question on tangent space and jet spaces

    Hello, I am reading some material related to jet spaces, which at first glance seem to be a generalization of the concept of tangent space. I am confused about what is the correct definition of a jet space. In particular, given a map ##f: X \rightarrow Y## between two manifolds, what is the...
  9. K

    I Uniqueness of tangent space at a point

    How do you show that there can be only one tangent space at a given point of a manifold? Geometrically it's pretty obvious in 3 dimensions, as one notices that there can be only one tangent plane at a point. But how could we show that using equations?
  10. K

    I Constructing the Tangent Space to the Sphere: A Simple Case Study in Relativity

    While studying Relativity I decided to take over a concrete case. So I thought of (what I think is) the simplest case which is the Sphere ##S^2##. So I want to construct the tangent space to the sphere. I think for this I need to embbed it in ##R^3##. I worked out similar problems in the early...
  11. S

    I Definition of tangent space: why germs?

    I am reading "An introduction to manifolds" by Tu. He starts off in Chapter 1 by introducing some definitions on ##\mathbb{R}^n## that will carry across to general manifolds. In Chapter 1, 2.2, he defines germs of functions as a certain equivalence class of smooth functions ##C^\infty_p##. I...
  12. O

    Understanding Tangent Space of S2n+1 in Cn+1

    1. Let p be an arbitrary point on the unit sphere S2n+1 of Cn+1=R2n+2. Determine the tangent space TpS2n+1 and show that it contains an n-dimensional complex subspace of Cn+1Homework Equations3. It is easy to find tangent space of S1; it is only tangent vector field of S1. But what must do for...
  13. G

    A Mapping Tangent Space to Manifold - Questions & Answers

    Hi all, this might be a silly question, but I was curious. In Carroll's book, the author says that, in a manifold M , for any vector k in the tangent space T_p at a point p\in M , we can find a path x^{\mu}(\lambda) that passes through p which corresponds to the geodesic for that...
  14. sunrah

    General vectors and tangent space

    Given a scalar function g defined on a manifold and a curve f:λ -> xa, the change of the function along the curve is \frac{dg}{d\lambda} = \frac{dg}{dx^{a}}\frac{dx^{a}}{d\lambda} = T^{a}\frac{dg}{dx^{a}} where \frac{dx^{a}}{d\lambda} = T^{a} is the tangent to the curve. The argument that I...
  15. hideelo

    Proof of dimension of the tangent space

    I am attaching a picture of a proof from the book "general relativity" by wald. This is supposed to show that the tangent space of an n dimensional manifold is also n dimensional. I have two questions. In equation 2.2.3 couldn't the function be anything at a since the (x-a) term is 0? How is...
  16. D

    Vectors in Tangent Space to a Manifold Independent of Coordinate Chart

    In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator [itex] X[\itex] acting on some function [itex]f:M\rightarrow\mathbb{R}[\itex] at a point [itex]p\in M[\itex] (where [itex]M[\itex] is an...
  17. C

    Demonstrating Tangent Space Independence in Manifolds

    Hello I'm french so sorry for the mistake. If we have a manifold and a point p with a card (U, x) defined on on an open set U which contain p, of the manifold, we can defined the tangent space in p by the following equivalence relation : if we have 2 parametered curve : dfinded from...
  18. orion

    Yet more elementary questions about the tangent space

    I am trying to self-study some concepts in differential geometry to try to update my knowledge from the old-style index games to something more meaningful. I know that there are many threads that have in some way addressed this, but I am still not understanding it completely. I'm new to this...
  19. mnb96

    Tangent space on complex manifolds

    Hello, I understand the concepts of real differentiable manifold, tangent space, atlas, charts and all that stuff. Now I would like to know how those concepts generalize in the case of a complex manifold. First of all, what does a coordinate chart for a complex manifold look like? Is it a...
  20. L

    Tangent space in manifolds, how do we exactly define?

    If we have a manifold with a chart projected onto ##R^n## cartesian space and define a curve ##f(x^\mu(\lambda))=g(\lambda)## then we can write the identity \frac{dg}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial f}{\partial x^\mu} in the operator form: \frac{d}{d\lambda} =...
  21. M

    Confusion regarding differential forms and tangent space (Spivak,Calc. on Manifolds)

    I have been working through Spivak's fine book, but the part about differential forms and tangent spaces has left me confused. In particular, Spivak defines the Tangent Space \mathbb R^n_p of \mathbb R^n at the point p as the set of tuples (p,x),x\in\mathbb R^n. Afterwards, Vector fields are...
  22. V

    Do all four-vectors live in a tangent space?

    Working through intro GR at the moment and I'm a little unclear on how tangent spaces are used to carry four-vectors over from SR to GR. So, at every point in spacetime we construct a tangent space. We can form a basis for this space with the tangent vectors (i.e. the four-velocities) of one...
  23. M

    Tangent Space Definition (Munkres Analysis on Manifolds)

    Hi all, I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition: Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair...
  24. G

    Tangent space as best approximation

    Dear all, in what sense the tangent space is the best approximation of a manifold? The idea is clear to me when we think about a surface in Rn and its tangent plane at a point. But what does this mean when we are referring to very general manifolds? In what sense "approximation" and in what...
  25. K

    Tangent space and tangent plane

    Homework Statement So I'm a little confused about what a tangent space is. Is it the same as the equation of the tangent plane in lower dimensions? My notes define the tangent space as follows. Let M be a hypersurface of Rd. Let x(s) be a differentiable curve in M such that x(0)=x0 is in...
  26. Alesak

    Definition of tangent space on smooth manifolds

    Hi, I'm having trouble understanding why is tangent space at point p on a smooth manifold, not embedded in any ambient euclidean sapce, has to be defined as, for example, set of all directional derivatives at that point. To my understanding, the goal of defining tangent space is to provide...
  27. P

    Understanding Tangent Space Basis: Proving Intuitively

    I am unable to understand as to how the basis for the tangent space is \frac{\partial}{\partial x_{i}}. Can this be proved ,atleast intuitively? Bachman's Forms book says that if co-ordinates of a point "p" in plane P are (x,y), then \frac{d(x+t,y)}{dt}=\left\langle 1,0\right\rangle...
  28. D

    Notation for basis of tangent space of manifold

    I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it? for example, i know...
  29. E

    What is the Tangent Space for a Given Matrix A?

    Homework Statement Homework Equations The Attempt at a Solution
  30. T

    Tangent space vs. Vector space

    I'm not sure I fully understand the difference between these two terms when used in differential geometry/general relativity. If I were to describe covariant differentiation to someone, I would say something like this: "On a curved manifold (imagine a basketball), you could assume a tangent...
  31. W

    Why the basis of the tangent space of a manifold is some partials?

    it is quite peculiar i know you do not want to embed the manifold into a R^n Euclidean space but still it is too peculiar it is hard to develop some intuition
  32. K

    Tangent Space to Unitary Group

    This may seem like an easy question, but my differential geometry is a little rusty. I'm trying to find the tangent space to the Lie group U(n) ; that is, for an arbitrary X \in U(n) I'm trying to find an expression for T_X U(n) . I can't quite remember how to do this. I've been playing...
  33. N

    Maximal Exterior product of Tangent space

    I am working through a book on Kahler manifolds and for one of the proofs it states that the maximum exterior power of TM is m (where M has complex dimension). Could you explain why this is the case rather than the maximum exterior power being 2m.
  34. B

    Why is the tangent space of a lie group manifold at the origin the lie algebra?

    Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me. I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this...
  35. G

    Basis for tangent space and cotangent space

    Hello, I'm trying (somewhat haphazardly) to teach myself about differential forms. A question I have which is confusing me at the moment is about the tangent and cotangent spaces. In https://www.physicsforums.com/showthread.php?t=2953" the basis for the tangent space was described in terms of...
  36. Q

    Proving \alpha^i = v x^i on tangent vector space T_pM"

    Let me first confess this a copy/paste of a question I asked on another forum; I trust it's not against the rules. Let M be a C^{\infty} manifold, and, for some neighbourhood U\ni p \subsetneq M let there be local coordinates x^i such that p=(x^1,\,x^2,...,x^n) Suppose that T_pM is a...
  37. J

    Inverse in lie group, tangent space

    Homework Statement I'm supposed to prove, that when G is a Lie group, i:G\to G is the inverse mapping i(g)=g^{-1}, then i_{*e} v = -v\quad\quad\forall \; v\in T_e G where i_{*e}:T_e G \to T_e G is the tangent mapping. Homework Equations I'm not sure how standard the tangent mapping...
  38. M

    Tangent Space of Singel layered hyperboloid

    Tangent space of single layered hyperboloid Ok i´m given a single layered hyperboloid given by \left(\frac{x}{a}-\frac{z}{c}\right)\cdot\left(\frac{x}{a}+\frac{z}{c}\right)-\left(1-\frac{y}{b}\right)\cdot\left(1+\frac{y}{b}\right)=0 Now the Problem asks me to take this as a vanishing...
  39. M

    Tangent space, derivation defifinition

    I'm reading "tensor analysis on manifolds" by Bishop and Goldberg. I have taking a course in differential geometry i R^3. The course was held on Do Carmos book. Do carmo deffined the tangent at a point on a surface as all tangents to all curves on the surface going through that point (or...
  40. A

    How Do You Determine Basis Vectors for the Tangent Space of a Surface in R^4?

    Stuck with problem: There is a local surface \alpha(u) = (f_1(u), f_2(u), f_3(u), f_4(u)) \in R^4. I need to find basis vectors of tangent space on that surface in some point p. It is not difficult to calculate tangent space for that surface, we should choose some curve on the surface and...
  41. D

    Finding the Tangent Space of SL(n,real) with A(0) being the Identity Matrix

    Hi, I am trying to find the tangent space of SL(n,real) where A(0) is defined to be the identity matrix. First of all I worked on the case when n=2 and found that the tangent space was A = \left( \begin{array}{ccc} a & b \\ c & -a \end{array} \right) where a,b,c belong to the...