The following Picture is the proof of "distance function on Riemannian manifold is a metric".
How the author deduced that length of any curve from $p$ to the boundary of given ball is at least $r/\lambda_0$?
Thanks.
2026-05-15 08:36:02.1778834162
A question on the proof of induced metric from Riemannian metric
157 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 METRIC-SPACES
- Show that $d:\mathbb{C}\times\mathbb{C}\rightarrow[0,\infty[$ is a metric on $\mathbb{C}$.
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Lemma 1.8.2 - Convex Bodies: The Brunn-Minkowski Theory
- Closure and Subsets of Normed Vector Spaces
- Is the following set open/closed/compact in the metric space?
- Triangle inequality for metric space where the metric is angles between vectors
- continuous surjective function from $n$-sphere to unit interval
- Show that $f$ with $f(\overline{x})=0$ is continuous for every $\overline{x}\in[0,1]$.
- Help in understanding proof of Heine-Borel Theorem from Simmons
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
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
geometry
circles
algebraic-number-theory
functions
real-analysis
elementary-set-theory
proof-verification
proof-writing
number-theory
elementary-number-theory
puzzle
game-theory
calculus
multivariable-calculus
partial-derivative
complex-analysis
logic
set-theory
second-order-logic
homotopy-theory
winding-number
ordinary-differential-equations
numerical-methods
derivatives
integration
definite-integrals
probability
limits
sequences-and-series
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?
Write down the integral for the length of any path from $p$ to the boundary of the ball (presumably they're choosing $r$ so that the point $q$ lies outside the ball). If you have a symmetric matrix $A(x)$ whose smallest positive eigenvalue is at least $\lambda_0$, what can you say about $\sqrt{u^\top A(x) u}$ for any vector $u$? Now integrate the appropriate expression to get the length of the path, and what's the lower bound you have?