- #1
Bowles
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What Lie groups are also Riemann manifolds?
thanks
thanks
Lie groups as riemann manifolds
- Thread starter Bowles
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- Groups Lie groups Manifolds Riemann
In summary, Lie groups can have Riemannian structures by assigning a metric to its tangent spaces, although this is only possible for abelian Lie groups. There can be many possible metrics assigned to a smooth manifold, and the group SO(n) can be understood as a projective n-space or n-sphere, but it is non-abelian and therefore cannot have a metric connection. The term "bi-invariant Riemann metric" refers to a metric that is preserved by both left and right multiplication, and for compact groups it is the same as a two-sided invariant metric. Finally, John Baez's definition of classical Lie groups uses conjugate transpose.
- #2
DeadWolfe
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Well any smooth manifold admits a Riemannian structure. For a lie group one could give it a metric by specifying the value at the identity component and extend to other tangent spaces by pushing forward along the left multiplication map, that is, by defining the metric to be left invariant.
- #3
Bowles
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Thanks DeadWolfe, that helped.
Follow-up question 1: giving a smooth manifold are there many Riemann structures we can assign or just one? I assume the later.
Question 2: sometimes I read the rotation groups SO(n) are diffeomorphic to the n-sphere with antipodal points identified, sometimes that they are groups of isometries that preserve the metric on an n-sphere. How does that go together?
thank you
Follow-up question 1: giving a smooth manifold are there many Riemann structures we can assign or just one? I assume the later.
Question 2: sometimes I read the rotation groups SO(n) are diffeomorphic to the n-sphere with antipodal points identified, sometimes that they are groups of isometries that preserve the metric on an n-sphere. How does that go together?
thank you
- #4
OrderOfThings
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Bowles said:
If you mean in the sense that the inherit connection can be given as a metric: This is only possible for abelian Lie groups and will turn such a group into a Euclidean space.
- #5
OrderOfThings
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1: There are many possible metrics that can be assigned to a smooth manifold. Although a sphere and an ellipsoid are equivalent as smooth manifolds, the geodesics are very different. The problem of calculating geodesics of an ellipsoid is quite interesting, it turns out that the geodesics are integrable due to "hidden symmetries".
2: The definition of SO(3) is the set of 3x3 matrices (actually a representation) preserving the distance to the origin and not reversing orientation. These are the ortogonal matrices with unit determinant. But a 3x3 matrix can also be thought of as a point in [itex]R^9[/itex]. The SO(3)-matrices form a 3-dimensional submanifold in this space and this manifold is [itex]P^3[/itex].
2: The definition of SO(3) is the set of 3x3 matrices (actually a representation) preserving the distance to the origin and not reversing orientation. These are the ortogonal matrices with unit determinant. But a 3x3 matrix can also be thought of as a point in [itex]R^9[/itex]. The SO(3)-matrices form a 3-dimensional submanifold in this space and this manifold is [itex]P^3[/itex].
- #6
Bowles
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Thanks OrderofThings!
Answer 1 makes perfect sense to me.
But for 2 I have to further ask: when SO(n) can be understood as projective n-space or the n-sphere why are you saying only abelian Lie groups can be given a metric? SO(n) is non-abelian.
Answer 1 makes perfect sense to me.
But for 2 I have to further ask: when SO(n) can be understood as projective n-space or the n-sphere why are you saying only abelian Lie groups can be given a metric? SO(n) is non-abelian.
- #7
OrderOfThings
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The smooth manifold P³ can be assigned different connections. One possibility is an isotropic constant-curvature torsion-free connection. This is a metric connection and makes P³ look locally as a 3-sphere. Another possibility is an isotropic no-curvature connection with torsion. This is the one that makes P³ into a Lie group. Since the connection has torsion it cannot be stated as a metric. The torsion manifests itself as non-commutativity of the Lie group.
Only abelian Lie groups have connections with no curvature and no torsion. Having no torsion they can be stated as metric connections (but quite dull such).
Only abelian Lie groups have connections with no curvature and no torsion. Having no torsion they can be stated as metric connections (but quite dull such).
- #8
Bowles
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Cool, that's some great information!
Many thanks again, OrderOfThings.
Many thanks again, OrderOfThings.
- #9
Bowles
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One more thing, though.
While google searching, I saw the term 'bi-invariant Riemann metric on Lie groups' quite often. What does it mean? Is it different from the 'ordinary' Riemann metric', because it seems it applies to all compact, semi-simple Lie groups?
While google searching, I saw the term 'bi-invariant Riemann metric on Lie groups' quite often. What does it mean? Is it different from the 'ordinary' Riemann metric', because it seems it applies to all compact, semi-simple Lie groups?
- #10
Bowles
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Still love to know what 'bi-invariant Riemann metric on Lie groups' means and how it corresponds to what OrderOfThings wrote so far.
Any input is welcome.
Any input is welcome.
- #11
OrderOfThings
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An invariant metric on a Lie group is a scalar product on the tangent spaces that is preserved when one moves around in the group (i.e. parallel transport using the inherit connection). Such a metric is very easy to construct: Choose a scalar product at the identity and define the scalar product at any other point to be the result of transporting the vectors to the identity and calculating the scalar product there.
The transport of tangent vectors can be done using either left or right multiplication and consequently the metric will be either left- or right-invariant. For compact groups it doesn't matter and the metric will be bi-invariant (also called two-sided invariant).
The transport of tangent vectors can be done using either left or right multiplication and consequently the metric will be either left- or right-invariant. For compact groups it doesn't matter and the metric will be bi-invariant (also called two-sided invariant).
- #12
Bowles
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You the man, OrderOfThings. All what you wrote here went straight in my notebook.
I honestly hope you get paid for your knowledge.
Now a complete different things, but which is not worth a own thread.
John Baez defines http://math.ucr.edu/home/baez/octonions/node13.html" the classical Lie groups. But why xx*=1 (conjugate), shouldn't it be the transpose?
I honestly hope you get paid for your knowledge.
Now a complete different things, but which is not worth a own thread.
John Baez defines http://math.ucr.edu/home/baez/octonions/node13.html" the classical Lie groups. But why xx*=1 (conjugate), shouldn't it be the transpose?
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- #13
OrderOfThings
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The joy of others is my award.Bowles said:
Yes, that is conjugate transpose. For real matrices it reduces to ordinary transpose.
1. What is a Lie group?
A Lie group is a mathematical concept that combines the properties of a group (a set of elements that can be combined in a specific way) and a smooth manifold (a space that is locally Euclidean, meaning it can be approximated by flat spaces).
2. How are Lie groups related to Riemann manifolds?
Lie groups can be thought of as a special type of Riemann manifold, where the group operations are smooth and the group elements can be continuously varied. This allows for the application of differential geometry techniques to study the structure of Lie groups.
3. What are some examples of Lie groups?
Examples of Lie groups include the rotation group in three dimensions (SO(3)), the special linear group (SL(2,R)), and the general linear group (GL(n,R)). Lie groups also have applications in physics, such as the Lorentz group in special relativity and the symmetry groups in quantum mechanics.
4. How are Lie groups used in mathematics?
Lie groups have many applications in mathematics, including differential geometry, representation theory, and algebraic topology. They are also used in physics and engineering, particularly in the study of symmetries and transformations.
5. What is the significance of Lie groups as Riemann manifolds?
Lie groups as Riemann manifolds have a unique structure that allows for the application of differential geometry techniques, making them useful for studying symmetry and other mathematical concepts. They also have important applications in physics, particularly in the study of symmetries and transformations in physical systems.
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