## Geometry – Non-cutting arcs from the origin

I am looking for a general way to find the center and radius of a circle that intersects an origin and any point so that the arc between the points does not intersect any other arc of the same origin.

EDIT: Sorry, stupid question. Selecting a radius large enough to approximate a line for the arc between points satisfies the condition.

## Differential geometry – For a curve in 3-space, the tangent always intersects a fixed line, then the curve is planar

This is an old exam question where I do not know how to start.

$$begin {array} {l} { text {Let} alpha text {is a regular curve in} mathbb {R} ^ {3} text {. Prove that when a tangent line intersects with} alpha text {a solid}} \ { text {line} l in mathbb {R} ^ {3}, text {then} alpha text {is planar. }} end {array}$$

My first problem is how to actually use the main assumption. I would like to express the condition that these two lines intersect algebraically to deduce that the torsion is $$0$$, I parametrize $$l = a + b cdot t$$but then I do not know if I should get an expression for differentiating (or manipulating) that characterizes the assumption that the tangent and $$l$$ to cut. Any hints?

## Agal Algebraic Geometry – Variation of the Hodge structure of a complex surface and reference requirement

Imagine a complex, smooth surface that is triple embedded. I'm interested in the limit at which the surface degenerates into a reducible (but reduced) surface. In other words, I have a family of surfaces.

$$begin {eqnarray} pi: mathcal {S} rightarrow Delta, end {eqnarray}$$
Where $$Delta$$ is parameterized by a complex variable $$t$$, and $$S_t = pi ^ {- 1} (t)$$ is smooth for $$0 0$$ and, $$S_0$$ is the degenerate surface.

My questions are:

1- What kind of rank one coherent sheaf on $$S_0$$ may be the limit of a bundle of cables $$S_t$$, Is it true that any element of $$Pic (S_0)$$ are limit of the line bundles $$S_t$$?

2- Assume that I select trunk bundles over each component of $$S_0$$ they are not a limit of a line bundle, so it is equivalent $$(1.1)$$ drive in $$S_0$$, Then I deform myself $$S_0$$ to something smooth $$S_t$$What happens to these (1,1) cycles?

For some physical reasons, I expect the answer to the first question to be yes, and for the second I hope it should become one $$(2.0)$$ Cycle after deformation.

Please let me know if there are references to the above questions that are not too abstract (for a physics student to read).

Thanks.

## Riemannian geometry – connection between Yamabe invariant and Yamabe equation

I'm trying to understand Lee and Parker's solution to the Yamabe problem. It seems to me that the constant $$lambda$$ what appears in the Yamabe equation $$square varphi = lambda varphi ^ {p-1}$$ and the Yamabe invariant $$lambda = inf_ varphi Q_g ( varphi)$$ Where $$Q_g$$ is the functional $$Q_ {g} ( varphi) = frac { int_ {M} left (a | nabla varphi | ^ {2} + S varphi ^ {2} right) d V_ {g}} { | varphi | _ {p} ^ {2}} = frac {E ( varphi)} { | varphi | _ {p} ^ {2}}$$ are indeed the same, that is, if $$varphi$$ is an absolute minimum for the function $$Q_g$$ then $$varphi$$ is a solution to the Yamabe equation with $$lambda$$ as the coefficient of $$varphi ^ {p-1}$$, It seems to me that Aubin has explicitly demonstrated this in his book "Some Nonlinear Problems in Riemannian Geometry."

An explicit calculation of the Euler-Lagrange equation for $$Q_g$$ (This is shown on page 39 of this paper by Lee and Parker and can easily be done explicitly.) This shows that the Euler equation is
$$a Delta varphi + S varphi | varphi | _ {p} ^ {- p} E ( varphi) varphi ^ {p-1} = 0$$
that is the Yamabe equation with
$$lambda = frac {E ( varphi)} { | varphi | _ {p} ^ {p}}.$$
For the Yamabe invariant to be the same constant that occurs in the Yamabe equation, the exponent on the denominator should be 2 and not $$p$$,
So I'm a bit confused: Is the Yamabe invariant the same constant in the Yamabe equation in the presence of a solution? $$varphi$$? If so, where do I go wrong?

## dg.differential geometry – The effect of the Hodge \$ star \$ operator on the symplectic structure of a Kahl \$ 4 \$ distribution

To let $$(M, omega, J, g)$$ be a $$4$$ Dimensional Kahler manifold. Put $$omega & # 39; = star omega$$ Where $$star$$ Is the hodge operator assigned to the metric? $$g$$,

is $$(M, omega & # 39;)$$ a symplectic manifold? Is it necessary symplectic equivalent to the original structure $$(M, omega)$$Is this a diffeomorphism? $$f$$ what carries $$omega & # 39;$$ to $$omega$$?

## dg.differential geometry – "Antipodal" maps on Riemannian manifolds

Inspired by this question, we ask the following question:

What is an example of a compact Riemannian manifold $$M$$ with diameter $$D$$ This is not ismoteric for a round sphere or projective spaces, but allows a diffeomorphism $$f$$ so that $$d (x, f (x)) = D, quad forall x in M ​​$$?

## Ag.algebraic geometry – angle between lines

I have three points in 2D, S, D and P. This gives the following pairs of numbers:
(Sx, Sy), (Dx, Dy) and (Px, Py)
Two lines are defined, SD and SP.

An algorithm must be developed (which must be implemented in SQL) to determine whether SP is clockwise or counterclockwise from SD.

## Ag.algebraic geometry – Spherical perverse sheaves on the affine Grassmannian and critically twisted \$ D \$ module

To let $$G$$ be a reductive algebraic group and leave $$Gr_G = G ((z)) / G ((z))$$ be his affectionate Grassmannian. Define $$mathcal {D} (Gr_G) _ {crit} -mod$$ to be the category of law $$D$$Modules $$Gr_G$$ twisted by the critical level of $$G$$ (which is $$-1 / 2$$ the killing form).

In this work by Frenkel-Gaitsgory is mentioned on page $$1347$$ that there is an equivalence
$$mathcal {D} (Gr_G) _ {crit} -mod ^ {G ((z)}} cong Rep (G ^ { vee})$$
between the category of $$G ((z))$$ Equivariant critically twisted $$D$$Modules $$Gr_G$$ and representations of the Langlands dual. I'm a bit confused because geometric Satake equivalency is the most commonly used $$G ((z))$$equivariant perverse sheaves on $$Gr_G$$, and not $$D$$Modules.

From Riemann-Hilbert I know that perverse sheaves are related to regular holonomic sheaves $$D$$Modules, but I do not understand how this gives us the equivalence of the categories written above. Does anyone have a quick explanation for that? Thank you in advance.

## Agal Algebraic Geometry – Is it possible to embed a group schema into a locally constant one so that the quotient exists?

To let $$S$$ be a sufficiently good basic scheme (such as the finite type over an algebraic closed field) and $$G to S$$ be a flat group scheme. I would like to ask: Can we always find a closed embedding? $$G to H$$ into another flat group scheme $$H$$, so that $$H$$ is constant over S locally in the Zariski topology, and that $$H / G$$ exists as a schema?

## dg.differential geometry – expressing the Riemann metric as a pullback metric

For a Riemannian manifold $$M$$ with an original metric $$g$$for any other metric $$g & # 39;$$ on $$M$$ Is there a diffeomorphism? $$f: (M, g) rightarrow (M, g)$$ so that the pullback metric of $$f$$ is $$g & # 39;$$? If this is not generally true, what kind of metrics apply, and which properties make it generally inappropriate? I am especially interested in the simple case in which $$M$$ is $$mathbb {R} ^ n$$ and $$g$$ is the usual Euclidean metric, but I'm also interested in the general case.