Now there are spaces called symmetric spaces that can roughly have whatever holonomy you like. They're also completely classified, so from the perspective of holonomy, maybe not so interesting.
So, question. I'm on a simply connected manifold, that's not locally a symmetric space, and such that the holonomy is not reducible. What can the holonomy be?
It can also be the unitary group $U(n) \subset SO(2n)$. This happens for the so-called Kahler manifolds; complex manifolds equipped with a "compatible" Riemannian metric.
(Given a metric $g$, the existence of a complex structure compatible with the metric is equivalent to having holonomy contained in $U(n)$.)
There's $SU(n)$. These are the so-called Calabi-Yau manifolds. You might think of them as being complex manifolds that are "complex orientable"; given a Kahler manifold, you can modify the metric so that it has holonomy $SU(n)$ but is still compatible with the complex structure iff the first Chern class $c_1=0$.
(Roughly, if its Ricci curvature vanishes in de Rham cohomology, then you can change the metric to make the Ricci curvature vanish identically. This is the so-called Calabi conjecture, proved by Yau, hence the name Calabi-Yau manifold.)
@JoshuaA As Tobias said, that's an unusual intro to alg varieties, because historically the study of zero locus of polynomials came waaaaa... (20 million a's) ... aay before even the word "sheaf" was spelled in math.
Next on the list is $\text{Sp}(n) \cdot \text{Sp}(1)$. This is a structure on $4n$-manifolds. (I never quite got what that gorup is supposed to be. The cdot isn't a literal product.) It corresponds to "quaternionic Kahler manifolds".
Then $\text{Sp}(n)$; these are known as hyperkahler manifolds. They stand to quaternionic Kahler manifolds as Calabi-Yaus stand to Kahler manifolds. These are popular but maybe a bit hard. Actually (other than slight deformations of the metric), there's only two of these known in most dimensions $4n>4$.
I think recently we now have new examples of them in dimensions 12 and 20 but that's it.
Popular despite us not having like any examples.
OK, so now I've given you two manifolds with "real structures" - one $O(n)$, the other $SO(n)$. Similarly, I've given you two kinds of holonomy that talks about a complex structure and two about a quaternionic structure, one "orientable", one not.
I don't know, but this discussion has already given me a burst of motivation for studying the topics from my course, which can be rather dull sometimes
So these things are mysterious but interesting geometries. We have examples of compact maniolds with this kind of holonomy due to Joyce. But so far only finitely many smooth manifolds up to diffeo are known to support a G_2 structure.
It doesn't help that we have very few tools to tell G_2-manifolds apart. There is a program, due to Donaldson (actually he writes a new paper every 10 years saying "hey guys could you just do this thing now?"), to do gauge theory (like the 4-manifold theory that helps you tell apart smooth structures) on G_2-manifolds, which should help us tell their deformation types apart.
This is still in the stages where people are doing the requisite analysis. No actual invariants have been defined. But it's a cool, hot topic.
The same sort of invariants tie in with Spin(7) manifolds and SU(3)-manifolds.
Probably not on special holonomy groups, but there's a good chance you'll learn some stuff about holonomy in the broad. I'd be glad to suggest references when your course finishes.
A variant of the local degree formula tells us that we can compute the degree of $f$ as follows: Pick any regular value and sum over the signs of the determinant of the Jacobian at each point in the pre-image of the regular value. That number is the degree
Ok, yes, if I choose an orientation chart $\phi : B \to \Bbb R^n$ around $x$, then it's also an orientation chart around $y \in B$. And then by definition $\phi^{-1}$ sends $u$ of $H_n(\Bbb R^n, \Bbb R^n - 0)$ to $\mu_x$ in $H_n(M, M - x)$ and $\mu_y$ in $H_n(M, M - y)$.
You keep those chart $h:B\subset M\to \Bbb R^n$ for which $u$ corresponds to $\mu_x$ under the iso $$(h^{-1})_*:H_n(\Bbb R^n, \Bbb R^n-0)\to H_n(B,B-x)\cong H_n(M,M-x)$$ If the chart $h$ does not satisfy that, then you take $\text{reflection} \circ h$
@iwriteonbananas Yeah. But what I was really hunting for (which I don't know if exists) is a direct relation between tangent spaces and local homology groups. In the proof, we just worked at the level of charts, not at the level of tangent spaces (or so I interpreted it)
I'd want to know how much we can mimick the tangent space using the orientation sheaf.
(the ambitious question is, "is there a topological tangent space?")
Hello guys, I recently read through my notes and it has a part that says if V1, V2 are linearly independent then span(V1) is not a subspace of span(V1, V2) which seems absurd to me
Huh that's nonsense. Any element of your vector space which can be written as linear combination of elements of V_1 can also be written as linear combination of elements of V_1 and V_2.
So span(V_1) is a subset of span(V_1, V_2) regardless of V_1 and V_2 being linearly independent.
@UserX I mean, if V_1 and V_2 are linearly dependent, you can say span(V_1) = span(V_1, V_2). But V_1 and V_2 might as well be linearly independent bases of your vector space, in which case span(V_1) = span(V_1, V_2) too. So there shouldn't be a sensible characterization or anything.
I am given the following set of equations from a physics course, which is about longitudinal waves in rods.
My questions are:
On the second line you have $ (\frac{\partial \Delta}{\partial x})dx $
If you are already specifying you are doing a partial differentiation with respect to the x-dir...
@UserX span(v_1) not being a subset of span(v_1, v_2) is still nonsense. And if v_1, v_2 are linearly independent, what you can say is that span(v_1) is a strict subset of span(v_1, v_2).
That is, span(v_1) is not equal to span(v_1, v_2).
(Since, if it was, then for some c, av_1 + bv_2 = cv_1 for any a, b. i.e., (a - c)v_1 = bv_2. But then v_1, v_2 are independent, so that's impossible)
You can take a collection of vectors and write down it's span. Naturally, I assumed V_1 is a subset of the ambient vector space because of capitalized V.
I am given the following set of equations from a physics course, which is about longitudinal waves in rods.
My questions are:
On the second line you have $ (\frac{\partial \Delta}{\partial x})dx $
If you are already specifying you are doing a partial differentiation with respect to the x-dir...
That's a vague question, in my opinion. Mutation is a biological event, like any other event on Earth. So it's needed or not needed depending upon the definition of need.
Darwin clearly stated why species struggle to survive in his original work of The Origin of Species, but since then biology just takes it as "obvious".
Well, as to that question, population growth of any species is a geometric progression whereas amount of food on the planet grows arithmetically. So naturally one needs to fight to survive.
But I do not see why that has anything to do with mutation.
I can't parse that statement. Mutation need not enrich an organism with nature-favored qualities. What happens is that ones who do not get nature-favored qualities via mutation gets cancelled off - they die. Thus, by exclusion, the ones which remain have nature-favored qualities (iterate this through a few progenies).
[cont.] It's a common misconception that an organism in unfavorable conditions mutate to give rise to qualities which enable them to survive. It need not. Mutation is a pretty random thing which happens frequently in less developed organisms. But it is by exclusion that the strong ones survive. (this is, btw, where a random walk like structure comes in)
We have that $|x_{mi}| \to |x_i| \forall i=1, \dots, n$ with $\sum_{j=1}^n |x_{mj}|^2 \leq 1$. Do we get from that that $\sum_{j=1}^n |x_j|^2 \leq 1$ ? Does anyone have an idea?
@Semiclassical So you are the assistant of the prof? In which subject? @Semiclassical
I am given the following set of equations from a physics course, which is about longitudinal waves in rods.
My questions are:
On the second line you have $ (\frac{\partial \Delta}{\partial x})dx $
If you are already specifying you are doing a partial differentiation with respect to the x-dir...
