Let $M$ be a complete Riemannian manifold. Its segment domain is defined by: $$ \mathbf{seg}(p)= \{v\in T_pM: \exp_p(tv):[0,1] \to M \textit{ is a segment} \ \ \} $$ (Note: "segment" has many different terminology like "minimizing curve" and "minimizing geodesic segment") My question is: How to show $\mathbf{seg}(p)$ is closed in $T_pM$?
2026-03-28 10:35:56.1774694156
Are segment domains closed?
96 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in DIFFERENTIAL-GEOMETRY
- Smooth Principal Bundle from continuous transition functions?
- Compute Thom and Euler class
- Holonomy bundle is a covering space
- Alternative definition for characteristic foliation of a surface
- Studying regular space curves when restricted to two differentiable functions
- What kind of curvature does a cylinder have?
- A new type of curvature multivector for surfaces?
- Regular surfaces with boundary and $C^1$ domains
- Show that two isometries induce the same linear mapping
- geodesic of infinite length without self-intersections
Related Questions in RIEMANNIAN-GEOMETRY
- What is the correct formula for the Ricci curvature of a warped manifold?
- How to show that extension of linear connection commutes with contraction.
- geodesic of infinite length without self-intersections
- Levi-Civita-connection of an embedded submanifold is induced by the orthogonal projection of the Levi-Civita-connection of the original manifold
- Geodesically convex neighborhoods
- The induced Riemannian metric is not smooth on the diagonal
- Intrinsic vs. Extrinsic notions of Harmonic maps.
- Equivalence of different "balls" in Riemannian manifold.
- Why is the index of a harmonic map finite?
- A closed manifold of negative Ricci curvature has no conformal vector fields
Related Questions in GEODESIC
- Length of geodesic line equals distance between two points?
- What's the relation between the Darboux Frame and the Frenet-Serret on a oriented surface?
- Projection from an ellipsoid onto a sphere that preserves geodesy?
- Vector field on a geodesic
- Geodesic lines of the form f(at+b)
- How to actually find a minimizing path on a manifold?
- Calculating the round metric on $S^n$
- Geodesic equation on a codimension 1 submanifold of $\mathbb{R^{n+1}}$
- How can you numerically approximate the geodesic midpoint of 2 points on an ellipsoids?
- Compute geodesic circles on a Surface
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Remember that a curve is a segment if its length is equal to the distance between the endpoints; so
$$ \mathbf{seg}(p) = \{ v \in T_p M : d(p, \exp_p v) - |v| = 0 \} = f^{-1}(0)$$
where $f : T_p M \to \mathbb R$ is defined by $$f(v) = d(p, \exp_p v) - |v|.$$
Since the exponential map, the distance function and the norm are all continuous, $f$ is continuous; so the segment domain is the pre-image by a continuous map of a closed set, and is thus closed.